lacunas for hyperbolic differential operators with...

LACUNAS FOR HYPERBOLIC DIFFERENTIAL OPERATORS

WITH CONSTANT COEFFICIENTS I

BY

M. F. ATIYAIt, R. BOTT, L. G/~RDING

Introduction

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The theory of lacunas for hyperbolic differential operators was created by I. G.

Petrovsky who published the basic paper of the subject in 1945(1). Although its results

are very clear, the paper is difficult reading and has so far not lead to studies of the same

scope. We shall clarify and generalize Petrovsky's theory.

The various kinds of linear free wave propagation that occur in the mathematical

models of classical physics are of the following general type. There is an elastic ( n - 1 ) -

dimensional medium whose deviation from rest position is described by a function u(x)

with values in some R N and defined in some open subset 0 of R ~, one of the coordinates

being time. When there are no exterior forces, u satisfies a system of N linear partial

differential equations P(x, ~/~x)u(x)=0 where P is a linear partial differential operator

with smooth matrix-valued coefficients. Further, a unit impulse applied at some point y

produces close to y a movement that propagates with a locally bounded velocity in all

directions. This movement is smooth outside a system of possibly criss-crossing wave

fronts and vanishes outside the cone of propagation K(P, y), a conical region with its

vertex at y and bounded by the fastest fronts. Mathematically, the movement is described

by a distribution E = E(P, x, y) which is defined when x is close to y, vanishes when x is

outside K(P, y) and satisfies

P(x, ~/~x) E(P, x, y) = 6 (x -y )

so that E is a (right) fundamental solution of P. Under these circumstances we say that P

is a hyperbolic operator. Briefly, P is hyperbolic if it has a fundamental solution with

(1) See the references.

110 M . F . ATIYAH, R. BOTT AND L. GARDING

conical support as described above. When P =P(~/~x) has constant coefficients, its funda-

mental solution E is defined for all x, y, it is unique and depends only on the difference

x - y. I t will then be denoted by E(P, x) and we have

P(~/~x) E(P, x) = E(P, x)P(~/~x) = ~(x).

The support of E is contained in a closed cone with its vertex at the origin which is proper

in the sense that , apart from its vertex, it is contained in an open half-space xv ~ = xlv~ 1 §

�9 .. § > 0 where 0 ~:v ~ E RL We let hypN (0) be the class of these operators and write it as

hyp (v ~) when N = 1 so tha t the operators are scalar. An easy argument (Lemma 3.2)

shows tha t PEhypN (v~) if and only if det P E h y p (v~). The scalar operators are basic. Let

hyp (v~, m) be all P Ehyp (v~) of order m and let Hyp (v~) and Hyp (v~, m) be the homogeneous

operators in hyp (v~) and hyp (v ~, m) respectively. The operators P in hyp (v~) can be charac-

terized algebraically. Writing P = P(D) where D = a/i~x is the imaginary gradient, we at tach

to P and its principal part a their characteristic polynomials P(~) and a(~), ~ E C ~. Then

P E hyp (v ~) if and only if a(v~) 4 0 and P(~ + tO) ~:0 for all real ~ when ]Im t] is large enough.

The elements of hyp (v~) will also be considered as polynomials with this property. I t

follows tha t hyp (v~)=hyp (-v~). When P = a E H y p (~, m), the condition means tha t the

equation a(~ + tv ~) = 0 has m real roots t for every real ~. In particular, apart from a complex

constant factor, such an a(~) is real. I f the roots are all different except when ~ is propor-

tional to v~, a is said to be strongly hyperbolic and the corresponding class of operators

is denoted by H y p ~ (v~). When P E h y p (v ~) then a E H y p (0) but if a E H y p (v ~) then every P

with principal par t a is in hyp (v~) only if a E H y p ~ (v~). The class of P with a in H y p ~ (v ~)

will be denoted by hyp ~ (0) and its elements are also called strongly hyperbolic. A scalar

operator P(x, ~/~x) with variable coefficients turns out to be hyperbolic if the operators

x--+P(y, ~/~x) are strongly hyperbolic and their order is constant. We shall only t reat opera-

tors with constant coefficients and, apar t from some occasional remarks, also restrict

ourselves to scalar operators. The determinants of the non-scalar hyperbolic operators

(systems) tha t occur in mathematical physics are as a rule not strongly hyperbolic.

When P E h y p (v~), write P = a +b where a E H y p (v ~) is the principal par t of P so tha t

the degree of b is less than tha t of P. I t turns out tha t b cannot depend on more variables

than a. More precisely, if Q(~), ~ EZ = C ~, is a polynomial, let L(Q), the lineality of Q, be

the maximal linear space L such tha t Q is a polynomial on the quotient Z/L. When L(Q) = O,

Q is said to be complete. When P E h y p (~), P and a have the same lineality L(P)=L(a).

I t s eodimcnsion n(P) = n(a) is said to be the reduced dimension of P and a.

The class hyp (v~, 0) consists of all non-vanishing constants. For them, L ( P ) = Z .

The class hyp (v~, 1) consists of all P of degree 1 such tha t a(v~) =~0 and a(~) is real apar t

LACUNAS FOR H Y P E R B O L I C D I F F E R E I ~ T I ~ L OPERATORS 111

from a constant factor. In that ease L(P) is the hyperplane a (e)=0 and has dimension

n - 1 . With a suitable choice of coordinates, every complete a E H y p (v q, 2), n~>2, is a mul-

tiple of the wave polynomial A ( e ) = e ~ - e ~ - . . . - e ~ and then A(v q) >0. The corresponding

operator A=A(~/ax) is the wave operator. I t is strongly hyperbolic. When n =2 ; every

a E H y p (vq) is a product of real linear factors such that a(vq)4=0. When these factors

are all different, a E H y p ~ (vq).

By the algebraic definition of hyperbolieity, the classes hyp (vq) and Hyp (v q) are

closed under multiplication and factors of hyperbolic polynomials are hyperbolic. The

process of localization also leads to hyperbolic polynomials. When P(e) is a polynomial

of degree m, let us develop tmP(t-le + ~) in ascending powers of t and let tPP~(~) be the first

nonvanishing term. The integer p =m~(P) is called the multiplicity of e relative to P and

the polynomial $-~P~($) the localization of P at e. When P = a is homogeneous, a(e + t~) =

t'a~(~) +higher terms and p=m~(a) is simply the degree of the polynomial $-~a~(~). I t

turns out that if P E hyp (v~) and e is real, then P~ E hyp (v ~) and a~ E Hyp (O) is the principal

part of P~. In particular, m~ (P)= m~ (a). When P E hyp ~ (v ~) is strongly hyperbolic, all P~

with e 4=0 are constant or first degree polynomials so that, accordingly, n(P~)=n(a~)=0 or 1.

When a(e) is a homogeneous polynomial, let A: a(e) = 0 be the associated hypersurfaee

in Z and let Re A be the real part of A. The lineality L(a) depends only on A and will also

be denoted by L(A). In particular, when x = (x~ .... , xn) belongs to the dual complex space

Z', let X: x 1 el + ' " Xn en = 0 be the associated hyperplane and Re X its real part. By the

algebraic definition of hyperbolicity, when a E Hyp (v~, m) the surface Re A intersects every

real straight line with direction v a in m real points. When a E Hyp ~ (v a, m) and the line misses

the origin, they are all different. The component F(A, v~) of Re Z - R e A that contains v~

is an open convex wedge whose edge is Re L(A). When P E h y p (v ~) has principal part a,

we also put F(P, O)= F(A, v~).

A basic property of hyperbolic polynomials is that P E h y p (v~) implies P E h y p 07)

for every ~ EF(P, v ~) =F(A, v~). More precisely, there is a convex open subset F I = FI(A, v~)

of F = F(A, 0) such that 81~1C 1~1 when s ~> 1 and F = [A sF1, s >0, and P(e+_i~)4=0 when

EF 1. The fundamental solution E(P, x) = E(P, z~, x) of P with support in xvq ~>0 is simply

an inverse Fourier-Laplace transform of p - l ,

[ ,

(1) E(P, O, x) = (2 )-nJP(e e,x(,-,,, de, ~ E F 1.

Here the right side is independent of ~] and, by the Paley-Wiener-Schwartz theorem,

E vanishes at x unless x~ >~ 0 for all ~ E F. This condition defines the dual cone K = K ( P , v ~) =

112 M. F. ATIYAH, R. BOTT AND L. G/~RDING

K(A, z$) of F. This dual cone is the propagation cone for P relative to v ~. I t is closed and

convex and meets every half-space x~ ~< const, ~ E F, in a compact set. I t is also the closed

convex hull of the support of E. I t is contained in and spans the orthogonal complement

of Re L(A) whose dimension is the reduced dimension n(a)=n(P). More precisely, put

= (~', ~") where ~', ~ are coordinates in Z/L(A) and L(A) and let x', x" be the dual co-

ordinates in Z' . Then P'(~')=P(~) is complete, P(D)=P'(D')| 1 where 1 is the identi ty

operator on L(A), and (1) shows that E(P, ~, x)= E(P', ~', x')| (~(x"). This reduces the s tudy

of fundamental solutions to those of complete operators. Putt ing P = a + b we also have

(2) E(P, ~, x) - ~_ - (-1)kb(D)~E(a~+Iz$,x) 0

so tha t there is a further reduction to the homogeneous case. (2) follows from (1) and the

fact tha t ba-l(~-is~) tends to zero uniformly in ~ when s-+ ~ (Leif Svensson (1969)).

There is a simple connection between the fundamental solution E(x)= E(P, ~, x) of

P Ehy p (v~) and the fundamental solution E~(x)= E(P~, z$, x) of a localization of P pointed

out to us by L. HSrmander. I f p=m~(P) is the multiplicity of ~ E R e Z and t - ~ , then

tm-~e~t~E(x) tends to Ei(x) in the distribution sense. As a consequence we have the im-

portant fact tha t

(3) O : ~ E R e Z o S E ~ c S S E ,

where S denotes support and SS singular support.

Let us now consider the local cones F~=F(P~, v~)=F(A~, v~) and local propagation

cones K~=K(P~, v~)=K(A~,v~) where a~ is the localization of the principal par t of P.

I t is easy to show tha t F~ D F(A, v~), K~ c K(A, ~) with equality if and only if ~ E Re L(A). When a($)40, F~ is the whole space and K~=O. When a(~)=0 but grad a(~)~=0, then

F~ is a half-space and K~ is a half-ray normal to Re A. When a is strongly hyperbolic,

these exhaust all possiblities. At singular points of Re A outside the origin, Ff may be smaller

than a half-space and K f bigger than a half-ray. The local cones have important continuity

properties. The functions ~, a~F(A~, z~) and $, a~K(A~, ~) are inner and outer continuous

respectively. Here inner (outer) continuity of a map a from a space {T} to subsets of another

space means tha t a(v') N a(T), (a(3') U a(3)), is close to a(3) when 3' is close to 3.

Let ORe A be the real dual of the hypersurface Re A, i.e. the set of xf iRe Z' such tha t

Re X is tangent to Re A. I t has been known for a long time, in particular through the work

of Herglotz (1926-28) tha t if a E H y p (v~), E=E(a, ~, x) is holomorphic in /~(A, v ~) outside

~ A in significant special cases. For strongly hyperbolic a, the proof is due to Petrovsky.

There is also no better result. In fact, if aEI-Iyp ~ (v~), then, by (3), the singular support of

LACUNAS ~ OR H Y P E R B O L I C D~FFERENT~AL OPERATORS 113

E contains the support of every E$ with ~ =~0. Since the corresponding a~ is constant or

linear this support is all of K~=K(A~,O). Now K(A, O)N ORe A is the union of these

K~ and this shows tha t E is nowhere smooth across ORe A in K(A, z$). The same proof

works, also in the inhomogeneous case, when Re A is non-singular outside the origin.

When Re A is singular, ORe A is not the proper object to consider and we replace it by the

wave front surface W = W(A,~) defined as the union of all K(A~, ~) for 0 =4=~ real. I f a is not

eomplete, W=K=K(A, z$) but apart from tha t case, W has eodimension 1, is contained in

ORe A n K and contains the boundary of K. Unlike ORe A, W is an outer continuous func-

tion of a E Hyp (v~). I t also has the stronger continuity property tha t there are elements b

in H y p ~ (v ~, m) arbitrarily close to a given a such tha t W(B, ~) meets a given conical neigh-

bourhood of a given ray in W(A, ~). We shall show tha t E(P,v~, .) is holomorphie outside W(P,z$)=W(A,~) when

P E hyp (v ~) and a is the principal par t of P. When P = a, the proof is achieved by a shift

of the chain of integration in (1). The new chains are images of maps ~-+~-iv(~) where

v(~) is a smooth vector field such tha t v(~) E F~ (A, v ~) for all ~. By the inner continuity of

the function ~-~F~, such fields exist and we show tha t small multiples of v have the addi-

tional property tha t a(~-isv(~))40 when 0 < s < l . By Cauchy's theorem, the constant

vector field v0(~ ) = 7 in (1) can be replaced by any such v homotopic to v 0 provided the

exponential stays bounded. When x E 4- W(A, ~) ,we can in fact find a v with these proper-

ties such that , in addition, a(~-iv(~)) -1 is bounded and xv(~) < -e(~) for some e > 0 and all

large ~. But then we have absolute convergence in (1) and the integral is a holomorphio

function close to x. Applying this result and some simple estimates to (2) takes care of the

inhomogeneous case.

When P=aEHyp (v ~, m) is homogeneous, we can use the homogeneity to perform a

radial integration in (1). We perform this operation using our modified form of (1). The

vector fields tha t are adapted to this process constitute a family V= V(A, X, ~)charae. terized by the properties tha t v(~)EC ~ for 0 4 ~ real, v(2~)=i2iv(~ ) when 0~:~teR,

v(~)eF(A~,~)NReX for all ~ 0 , a(~-isv(~))40 for all real ~ 4 0 and 0<s~<l . When

x'E +W(A,O), this family is not empty and any two elements of it are homotopic.

Modulo constant factors the result (Theorem 7.16) may be presented as follows.

I f aE Hyp (v~, m) and xEK(A, ~) - W(A, v~), then

(4) (~/~x)" E(a, v a, x)'" 3 =f* (x~)q ~ a(~)-~ eo(~), q >1 O,

(5) (a/~x)'E(a'z~'x)'~ft, o~* (x~)q~a(~)-lo)(~), q < 0 .

8 -- 702909 Acta mathematica, 124. I m p r i m ~ lo 8 Av r i l 1970.

114 M . F . ATIYAH, R. BOTT AND L. G~LRDING

Here (~/~x)'= (~/aXl) "1 ..., ~ = ~1 ..., o~(~)= ~ ( - 1)J-l~jd~l ... d~j ... d~n and (4) and (5) hold

when q=m-n-Iv] >~0 and q < 0 respectively, I v] = ~ l + . . . + ~ n - T h e integrands are ra-

tional (n-1)- forms of homogeneity zero on Z and hence also closed forms of maximal

degree in (n - 1)-dimensional projective space Z* = ~/~ where a dot indicates that the origin

is removed. They are holomorphic on Z* - A * and Z* - A * U X* respectively. Here, if B is a

part of Z, B* denotes the image of ~ in Z*. The forms are integrated over certain homology

classes a*= a(A, x, v~) * and t~:r The class a* belongs to Hn_I(Z*-A*, X*) and is simply

the class of ~ where 2:% is the image of the map Re Z 3~--->~-iv(~), vE V(A, X, ~), and

is oriented by x~co(~)>0. Since V(A, X, z?) is one homotopy class, this defines a* uni-

quely. I ts boundary ~a*E Hn_,(X*-X*N A*) is an absolute class. The tube operation

t.: H~_~ (X* - X * N A*)-~ H~-I(Z* - A * U X*) is generated by the boundary of a small 2-disk

in the normal bundle of X* when its center moves on X*. Because of the orientation

x~co(~) > 0, the homology class ~* depends in a very essential way on the parity of n. When

aEHyp~ the formulas (4), (5) are essentially due to Herglotz and Petrovsky. The

formulas of Petrovsky are obtained by taking one residue onto A* in (4) and two succes-

sive residues onto A*N X* in (5). These operations are well-defined only if A* and

A* N X* are nonsingular. The class a* was introduced by Leray (1962).

A close study of the fundamental solutions of hyperbolic operators should involve

behaviour near the wave fronts and also determination of supports and singular supports.

The concept of a lacuna is basic in such a study at least for operators with constant coef-

ficients. Before going into the details we shall consider wave propagation in general.

As before, let u(x) be a distribution defined in some open part 0 of R ~ and think of it as

describing the movement of an elastic (n - 1)-dimensional medium. Let C(u) be the maximal

open part of 0 where u is a Coo-function. The complement of C(u) is then the singular

support of u, in our case the wave fronts. Let L be a component of C(u) and let x be a point

on the boundary ~L of L. We say that u is sharp from L at x if u has a C~176 from

L to L N M where M is some open neighbourhood of x. The physical implication of this

is clear. An observer that moves at will in space-time and studies the movement u in L

will not notice the part of the wave front at x before actually crossing it. Measurements

that he can make in L will not indicate the singular behaviour at x. In the contrary situa-

tion, of a non-sharp wave front, measurements in L close to x indicate the presence of a

front at x. Sharpness at all boundary points characterizes lacunas. More precisely, a

component L of C(u) is said to be a lacuna of u if u has a C~~ from L to L. When

u vanishes in L, L is said to be a strong lacuna. Petrovsky only considers strong lacunas.

The definition of lacuna given here follows a suggestion by L. H5rmander.

The lacunas for a hyperbolic operator P are simply the lacunas of the distributions

LACUNAS FOR HYPERBOLIC DIFFERENTIAL OPERATORS 115

x-~E(P, x, y), or, if P has constant coefficients, the lacunas of the distribution E ( P ) =

E(P, x). Consider the scalar case P E h y p (v~, m) and let a E H y p (v~, m) be the principal part

of P. We know that E is holomorphie outside the wave front surface W = W(A, v~). Hence,

if the singular support SSE of E fills all of W, all lacunas are components of the comple-

ment of W. This is the situation when Re A = A - ( 0 3 in regular, when n 4 3 and probably

also when n ~ 4. However, if n > 4, there are homogeneous operators a in Hyp (v ~) such that

W is not all of SSE(a). But we shall show that in all cases W = SSE(a k) when k is a large

enough integer. At present, we cannot give a complete description of SSE and SE for all P.

In any case it is convenient to extend the lacuna definition by allowing as lacunas for P

components L of the complement of W having the property that E(P) has a C~-extension

from L to L. Such lacunas are said to be regular. In particular, the complement of the

propagation cone K = K(P) is a regular strong lacuna, called the trivial lacuna. With this

definition, if x belongs to a regular lacuna L for P, then E(a) is a polynomial of homogeneity

m - n in L so that L is a (regular) lacuna for a. In fact, it is immediate from (1) that

t~-m(~/~x)VE(P, t$, tx)-~(~/ax)~E(a, v ~, x) in the distribution sense as t-~0. Keeping x in L,

the derivatives (~/~tx)VE(P,v~,tx) are supposed to have limits as t -~0 and hence

(~/ax)VE(a, v ~, x ) = 0 in L when ]v[ > m - n . By (1), E(a, v ~, x) has homogeneity m - n and

this finishes the argument. A similar reasoning gives the same result when P =P(x, ~/ax)

is hyperbolic with variable coefficients but as we shall stay with constant coefficients, we

do not give the details. In any case we have seen that a component of Re Z - W ( a , v ~)

is a lacuna for a E Hyp (v~) if and only if E(a, v a, �9 ) is a polya~omial there. Note that if m <n,

then E(a, v ~, .) has to vanish L so that L is a strong lacuna. If L is a regular lacuna not

only for a but for all powers of a and P E h y p (t$) has principal part a, then (2) can be used

to show that E(P, v ~, �9 ) is an entire function in L so that L is a lacuna for P.

The Herglotz-Petrovsky-Leray formulas (4), (5) immediately give sufficient conditions

for an a E H y p (v ~, m) to have non-trivial regular lacunas. In fact, by (5), if

(6) ~ ( A , x, v~)* = 0 in Hn_~(X* - X * N A*)

then x belongs to a regular lacuna L for all powers a ~ of a. If m l c - n < O , L is also a strong

lacuna. The condition (6) is essentially due to Petrovsky. For this reason, regular lacunas

L such that (6) holds for all x in L are said to be Petrovsky lacunas. The corresponding

condition connected with (4) and m ~>n, namely a * = 0 in Hn_I (Z* -A* , X*) would imply

that x belongs to a regular strong lacuna for a. This case is, however, illusory. In fact,

we shall prove in Part I I that if x is outside W but in the propagation cone, then a* has

non-zero intersection number with F(A, v~)*EH~_I(Z*-X*, A*) and hence cannot vanish.

All the classical lacunas are Petrovsky lacunas. If, e.g. n = 2 , then a is a product of

116 M . F . ATIYAH, R. BOTT AND L. G~RDING

linear factors and ~ * = 0 when x~- t -W so tha t the entire complement of W consists of

Petrovsky lacunas. 0n ly the trivial lacuna is strong. Here it is of course quite elementary

tha t E(a, v q, .) is a polynomial in each component of R e Z - W . I f n > 2 and a = A is the

wave operator and vq= (1, 0 ... . . 0) then the propagation cone K = K ( A , v a) is the forward

light-cone xl>~0, x ~ - ~ - . . . - x ~ >~0 and W = W(A, v ~) is its boundary. Here, if xE/~, ~ *

vanishes or not according as n is even or odd. This reflects the well-known fact tha t inside X2 X 2 ~(2 k - n)/2 K, E( Ak, v~, x)=Cn.k(x~ - 3 - . . . - ,~J wherec~ .k=0i f h i s even a n d 2 k < n a n d c ~ . k ~ : 0

otherwise. Hence K is a (regular) lacuna or not according as n is even or odd. (1) This fact,

sometimes called Huygens ' principle, was the starting point of the theory of lacunas. There

are examples of Petrovsky lacunas for every m and n. These lacunas are also stable in the

sense that they are not destroyed by small perturbations of a within the class H y p (v~, m).

In fact, the Pet rovsky condition (6) is stable under such perturbations. When A* is regular,

the proof is immediate. In the general case we have to use the continuity properties of

the local cones F(A~, v~). The continuity properties of the wave front surface W = W(A, v ~)

show that no lacuna L containing a piece of W can be stable under all hyperbolic perturba-

tions. In fact, there are operators b EHyp ~ (v a, m) such tha t W(B, v ~) and hence also the

singular support of E(b, ~, �9 ) comes arbitrarily close to any given ray in W.

I t seems fruitful to t ry to prove or perhaps disprove some simple but strong state-

ments about supports, singular supports and lacunas which are in agreement with known

facts. Let a E H y p (v ~, m), E ( a ) = E ( a , z$, .), K ( A ) = K ( A , v ~) and let a~EHyp (v ~) be the lo-

calizations. Our statements are

(i) all regular lacunas are Pet rovsky lacunas

(ii) SSE(a)= USE(a~) for 0 = ~ real

(iii) E(a) is holomorphic outside SSE(a).

The s ta tement

(iv) m >~no SE(a) =K(A)

turns out not to be true for certain operators a of the form b(~') c(~ ") where ~', ~" is a parti-

tion of the coordinates ~, but it would be enlightening to know precisely when it holds.

Pet rovsky proved a weak form of (i) tha t can be paraphrased as follows: all stable

lacunas for a strongly hyperbolic a with non-singular A* are Petrovsky lacunas. Doing this,

he used the topological machinery available at the time, in particular the results of Severi

and Lefschetz on the topology of algebraic manifolds. At first, requiring stability under all

perturbations of a and using the fact tha t E(a, v ~, x) is a holomorphie function also of a

when x is outside the wave front surface, he can work in a generic situation where all

(1) The explicit formulas of H a d a m a r d (1932) and M. Riesz (1948) show that this s ta tement also holds for wave operators with variable coefficients.

L A C U N A S F O R H Y P E R B O L I C D I F F E R E N T I A L O P E R A T O R S 117

deformations of A are permitted. One of his general results is the following one. Let h(~)

be a homogeneous non-zero polynomial of degree m - n ~> 0 and let q = ~0~ be the correspond-

ing hotomorphic (n -2 ) - fo rm on A*. Assume tha t for some fl E H,_2(A*) the integral Sp

vanishes considered as a function of a. Then fl =0. To prove this, Petrovsky analyses

H,~_2(A* ) explicitly. Assuming fl # 0 he is able to find an a with the property that Sp q #0.

The details involve a substantial amount of intuitive topology.

Working with the cohomology of Z * - A * and Z * - A * U X* instead of the homology

of A* and A* N X* as Petrovsky does, we shall come much closer to proving (i). This will

be done in Par t I I of our paper, but some of its results will be presented here. Our main

tool is a well-known theorem by Grothendieck (1966) generalizing an earlier result by Atiyah

and Hodge (1955). This theorem shows in particular tha t the cohomology of the complement

of an algebraic hypersurface in projective space is spanned by all rational forms with

poles on tha t hypersurface. I t has an immediate application to our case. In fact, the forms

tha t appear in (4), (5) applied to all powers of a constitute a basis for all rational ( n - 1 ) -

forms on Z* with poles on A* and A* U X* respectively. Since the kernel of tz is zero, this

shows tha t every regular lacuna for all powers of a is a Pet rovsky lacuna and tha t the

only strong regular lacuna for all powers of a is the trivial one. We shall also see tha t

Grothendieck's theorem holds with a bound on the order of the poles depending only on

m and n, so tha t our statements are true for all sufficiently high powers of a and hence

(i) holds in a weak form regardless of the singularities of A*. In this weak form and in

combination with (3) it implies tha t the wave front surface W ( A , v ~) is the singular support

of E(a ~, v ~, �9 ) for large enough k. All s tatements (i) to (iii) hold when n 4 3 or if A* is only

mildly singular. The statement (iv) is probably true when n ~<3 and it holds in the fol-

lowing weak form: when m>~n, a has no stable strong lacunas inside K.

In physics, hyperbolic systems are more important than the scalar operators which

have occupied us here. However, every component of the fundamental solution E(P, v a, �9 )

where P E HypN (0) is a sum of derivatives of E(det P, v ~, -). Hence it suffices to study

supports, singular supports and lacunas for scalar distributions of this form. For instance,

it seems desirable to show tha t if a E H y p (v~) and Q is relatively prime to a, then every

regular lacuna for Q ( D ) E ( a , v ~, .) is a Pet rovsky lacuna for a. This situation leads to

problems about the cohomology of rational forms tha t will be treated in par t I I .

This introduction combined with the table of contents should give a fair idea of the

problems we pose and our main results. A preliminary outhne has been published earlier

(Ghrding 1969), Finally, we should like to thank Lars HSrmander for his decisive contribu-

tions to our paper. We are also grateful to St Catherine's College, Oxford, under whose

aegis this s tudy was initiated.

118 M. F. ATIYAH, R. BOTT AND L. G~LRDING

Contents

Introduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Chapter 1. Distributions and the Fourier-Laplace transform

1. Generalities about distributions . . . . . . . . . . . . . . . . . . . . . . . 118

2. Inverse Fourier-Laplace transforms of functions holmorphic in tubes . . . . . . 124

Chapter 2. Fundamenta l solutions of hyperbolic operators with constant coefficients

3. Hyperbolic differential operators and hyperbolic polynomials . . . . . . . . . . 126

4. Fundamenta l solutions of scalar hyperbolic operators. Localization. Analytic con-

t inuat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5. The geometry of hyperbolic surfaces. Semi-continuity of the local cones. The wave

front surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6. Vector fields and cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7. Fundamenta l solutions expressed as rational integrals . . . . . . . . . . . . . 169

Chapter 3. Sharp fronts and lacunas

8. Supports and singular supports of fundamental solutions . . . . . . . . . . . . 179

9. Sharp fronts, lacunas, regular lacunas . . . . . . . . . . . . . . . . . . . . 182

10. Stability, Petrovsky lacunas . . . . . . . . . . . . . . . . . . . . . . . . 184

11. Remarks about systems . . . . . . . . . . . . . . . . . . . . . . . . . . 188

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Chapter 1. Distributions and the Fourier-Laplace transform

This chapter will present some general mater ia l to be used later on.

1. Generalities about distributions

Distributions on a mani/old. Let X be a paracompact C~-manifold of d imens ion n.

Let x be a po in t on X, (x 1 . . . . . xn) the coordinates of x a nd pu t

(0/ax)" = (0/0xl) ' , ... (~/0xn)'-,

where vl . . . . are integers /> 0. The order of this der ivat ive is lul = ~ 1 + ... +rn. Let C(X) =

Cr176 be the space of all complex C~~ on X a nd Co(K) the subspace of C(X)

whose elements have supports in a given compact subset K of X. Let Oo(X ) be the un ion

of all Co(K ). Topologize C(X) b y locally uni form convergence of all derivat ives a nd topo-

L A C U N A S I~OR H Y P E R B O L I C D I F F E R E N T I A L O P E R A T O R S 119

logize Co(X ) as the inductive limit of all Co(K ) with the topology induced by C(X) as K

runs through a sequence of compacts tending to X. Consider also n.forms on X,

~(x) =g(x)dx, d x = d x l A ... A dx,,

with densities g E C(X). Assume X to be oriented by some ~(x) > 0. Let ~(X), ~0(X), ~0(K)

be the spaces of forms 9 such tha t ~1~ belongs to C(X), C0(X ), Co(K ) respectively, tope-

logized in the obvious way. I f X is an open subset of R ~, we always take ~(x)= dx =

dxl A ... h dx,, The distributions on X are by definition the elements of the dual C'(X)

of ~0(X). The support of a distribution I will be denoted by S(I). We let C~(X) be the

space of distributions with compact supports. Every locally intcgrable function l(x)gives

rise to a distribution

We identify ! and this distribution. Also in the general case, distributions are writ ten as

](x), g(x) . . . . and we employ one or other of the two notations

(I, 9) = f xl(x) ~(~)

for the value of t at 9. I f X is an open subset of R n, we identify ~ with g(x)dx and write

(t, g) = I" l(x) g(~) dx. ,3x

Note tha t Co(X ) and hence also C(X) is dense in C'(X). The derivatives (O/Ox)'l(x) of a

distribution are defined in the domain V of the coordinates xl, ..., xn by

f (a/a~,)" l(x) g(x) dx = ( - 1)l'l f l(x) (alax)" g(x) dx,

where g(x)dxE~o(V ). Locally, a distribution is a finite sum of derivatives of continuous

functions. Let (x 1 ... . . xn) and (Yl . . . . , Yn) be coordinates with a common domain. The chain

rule of differentiation for functions,

~ll~y, = ~. (Oxk/ay,) ~11~ k

extends by continuity to distributions.

The singular support of a distribution of I is the complement of the maximal open

subset U of X where I equals a C~-function. I t will be denoted by SS(]).

120 M . F . ATIYAH, R. BOTT AND L. G ~ D I N G

Temperate distributions. The Fourier trans]orm. When g E C( Rn), put

Ig[N = sup (1+ [xl)N]~'g(x)[, x e R n, I~,] <N,

where [x I denotes some norm on R n, and let $ = $ ( R n) be all gEC(R n) for which all ]gin

are finite. Topologized by the norms ]giN, $ is a Fr~chet space, the space of temperate

test-functions, and its duM $ ' = S'(R n) is the space of temperate distributions (Schwartz

1950).

Let Z=(~, ~] .... } and Z'= (x, y .... } be two complex vector spaces of dimension n.

We shall always choose biorthogonal coordinates (~1, ...) and (x 1 .... ) so that x~ = Xl~ 1 + ... +

xn~ n and Ix [, I~1 will denote unspecified norms. The Fourier transform

= f dx, x, ~ real

is a linear homeomorphism $-~ $ with inverse

tha t extends to a linear homeomorphism $'-~ $', also denoted by :~.

When g E C 0 (Re Z'), its Fourier transform extends to the Fourier-Laplace transform

+ = f a(x) dx, '7 real,~

which is an entire holomorphic function. In connection with Fourier transforms it is often

convenient to use the imaginary gradient

D = i - ~ l ~ x

which has the property that ~:~ = :~D.

Distributions in one variable. Let X = R n and let / e C'(R). Then/(xl) can be considered

as an element of G'(X) defined by

where all forms are positive and g E Co(X ). The injection C ' (R)~ C'(X) obtained in this way

is eontinuous. When X is a manifold, there is a similar construction. Let s(x)EC(X) be

real and sueh that grad s(x)#O everywhere. The distribution /(s(x))EC'(X), /EC'(R),

is defined by


(1.1) fl(s(x))~(x)=~:(t)h(t)dt, where ~EE~0(X), V= R is the (open) range of s and

h(t) = f .x~_t~t(x ), ~0(x) =%(z) A dt

with V and s(x) = t oriented so tha t dt > 0 and pt(x) > 0 respectively, ~(x) > 0 being the orienta-

tion of X. In particular, if 1/9 O,

~J~ (s(z)) ~(z) ( - 1)~hc j~ (0),

where ~ is the Dirac measure on R. We note tha t also

h(t) = f / ( s ( x ) - t) q~(x).

For reference we state the following lemma whose proof is left to the reader.

1.2. L~MMA. The injection C'(R)~C'(X) defined by (1.1) is continuous.

Distributions which are boundary values o/holomorphic/unctions. Let A be an open

connected par t of the upper half-plane Yl > 0 of the complex (x 1 + iyl)-plane whose boundary

contains an interval I of the real axis and let B = R =-t be open. Let

/(x + iy) =/(x 1 + iy 1, x~ . . . . . xn)

be continuous in A • B and holomorphie in its first variable and assume tha t

(1.3) ](x+iy)=O(y~N), Yl 4 0

for some integer h r > 0, locally uniformly when x E I • B. Then

](x) = lim ](x +iy), y~ ~ 0

exists as a distribution, the limit being taken in the sense tha t

f [ ( x + iy) g(x) dx, yl ~ 0, (1.4) lim

exists for every g E Co(I • B). Moreover, if the limit vanishes, so does )r I n fact, let )tl,/~ . . . .

be successive integrals of / with respect to the first variable taken from a fixed point in A.

Then, using (1.3) one h a s / ~ ( x + i y ) = O(log y~l) and still another integration shows that ,

122 M.F. ATIYAH, R. BOTT AND L. Gi~RDING

as yl-~O, /N+l(x+iy) tends locally uniformly to a continuous function ]N+l(x) on I • B.

Now, by integrations by parts,

(1.5) f j ( X -J- ~y) g(X) dx = ( -- ] )N+l f /N+ 1 (X -~ ty) (0/OXl)N+I ~(Z) dx J J

and hence the limit (1.4) exists and defines a distr ibution/(x) = (O/~Xl)N+I/N+I(X) on I • B.

I f this distribution vanishes,/N+l(X) is a polynomial of degree <AZ+l in x 1 when x 2 . . . . are

fixed so that , by elementary function theory, /N+l(x+iy) is the same polynomial with

argument x 1 + i y 1 E A . Hence / vanishes in A • B.

In case A is a band 0 <y~ <eonst, B = R n-1 and (1.3) is replaced by

(1.3') /(x + iy) = O(y~ ~ (1 + Ix I) N)

then tN+~(x + iy) ~ o((1 + Ix I) ~'+1)

so tha t /(x) = (~/~xl)~+l/N+l(x)

is a temperate distribution. Note tha t in this case, (I.5) holds when g e t and tha t the

limit (1.4) exists also when g(x) =g~(x) depends on y provided g ~ g in $ when Y140.

Later on, in section 7, we shall employ certain explicit distributions on R which are

boundary values of functions analytic in the upper halfplane. We define them here and

state their main properties.

Let s, z be complex numbers and put

(1.6) Z s ( z ) = F ( - s ) e - ' ~ S z ~, sceO, 1,2 . . . . . 0 < arg z < ~ ,

Note tha t

(1.7) Re s < 0 * Zs(z) = i -~ Q-S-let~ arg i = g/2.

Hence, when Re s <0, gs is a constant times the Fourier-Laplace transform of a function

tha t vanishes on the negative axis and equals ~-s-1 on the positive axis. By the properties

of the gamma function,

(1.8) Z; (z) = gs_l(z).

Considered as a function of s, Z,(z) has simple poles at s = q = O , 1, 2 . . . . . We shah need the

first and second coefficients of the Laurent expansion of Zs, viz

(1.9) Zs+t(z) = -t-lz~ + Zs(z) +O(t),

L A C U N A S I~OR H Y P E R B O L I C D I F F E R E N T I A L O P E R A T O R S 123

where the first term vanishes unless s=q=O, 1, 2 . . . . . Since (1.5) does not define gs when

s =q, we are free to denote the constant term by gq in tha t case and it turns out to be con-

venient to do so. Both coefficients can be computed explicitly. By (1.6) and the properties

of the gamma function,

q )~q+t (Z) = - - H ( q - - k § t ) -11"~(1 - - $) e -n t t z q+t .

0 (1.10)

Hence

(1.11) z ~ (z) = zq /q! ; q = 0 , 1 , 2 . . . . .

X~ when s ~ 0 , 1 , 2 . . . . .

and, multiplying (1.9) by t and taking the derivative at t =0 , we get

(1.12) d

gq (Z) = -~ tXq + t ($) It = 0 = (q ! ) - 1 Zq (log z- 1 + cq + zd ),

q where c~ = F'(1) + ~ k -1, c o = F'(1).

1

I t follows from (1.12) that (1.8) holds also when s=0,1 .. . . and hence for all s. We note in

passing that (1.9) implies tha t

(1.13) d

tz.+, (z) l(t) I,=,, = z. (z) / (o) - , v . ~ (z) l ' (o)

when [ is holomorphic at the origin. In fact, in view of (1.9), both sides equal the constant

term in the Laurent expansion of tzs+t(z)/(t ) at t=0 .

In view of our earlier statements about boundary values of analytic functions, the

distributions

(1.14) Z~(X) =l img , (x+iy ) , y ~ 0, x E R

exist for all s. As functions of s, they have the same properties as before, in particular the

property (1.8). Passing to the limit in (1.12) we get

(1.15) q! Zq(X) = xq(log ] x [ -1 + cq) + 2-1~i(1 + sgn x) x q.

Hence, if aq E G'(R) is defined by

(1.16) 2~i(~q(x) =g~(x)- ( -1)qZ~(-~: ) , q=O, _+1 .....

the logarithm gets eliminated and we have

124 M. 1% ATIYAH, R. BOTT AND L. G~RDING

(~q(x) = 2-1(sgn x)xq/q! = 2-1(sgn x)i~~ q = 0, 1, ... :(1.17)

(rq(x) =~(-q-l~(x), q= - 1 , - 2 . . . . .

In fact, the first formula (1.17) follows from (1.15) and (1.16) and the second from the

first if we observe tha t (1.8) and (1.16) imply tha t (~'q =O'q_ 1.

2. Inverse Fourier-Laplace transforms of functions holomorphie in tubes

Duality o/cones. Let F c Re Z be a cone with its vertex at the origin, i.e. such tha t

2F = F when ~t > 0. Define the dual cone F ' c Re Z ' by

r ' = v er}.

Then F'= F' is a closed convex cone. In fact F'= H(F' ) where H denotes the closed convex

hull. Let F" = (~; ~x~>0, x E F ' } ~ R e Z

be the dual of F' . Since H(F) is the intersection of all closed halfspaces containing F, we

have F " = H ( F ) . Now, obviously,

so that , taking the dual of both sides,

(2.1) (r l n = H(r l +

I t is clear tha t F is contained in a hyperplane y~=O, y fixed in Re Z' , if and only if F ' ~ x

implies x-FtyEF' for all real t. Hence if F is convex, F is open if and only if F ' is proper

in the sense tha t r ' does not contain any straight lines. Let F be open and convex so tha t

F = H ( F ) . I t s dual K = F ' is then closed, convex and proper. Clearly, ~EF if and only if

~ x > 0 when x E I ~ = K - O . All ~ such tha t F + t ~ c F for all real t constitute a linear space

E, the edge of F. Obviously, K is contained in its orthogonal complement E ~ = {y; y E = 0}.

I t also spans this complement. In fact, if ~ K = 0 and ~EF, then ( ~ + t ~ ) / ~ = ~ / ~ > 0 for all

t so tha t ~ E E. I f E = 0, F is proper and K has a non-empty interior/~. One verifies tha t

x E K if and only if, for every real c, the half-spaces x~<~c have compact intersections

with F.

Inverse $'ourier-Laplace trans/orms. When g EC0(Re Z'), its Fourier-Laplace trans-

form

r e-tZ(~+~') g(x) dx + iT) = j

is an entire function and obvious estimates of

LACUNAS FOR HYPERBOLIC DIFFERENTTAL OPERATORS 125

(2.2) (~ + iT)v :~g(~ + iT) = f e -tx(~§ DVg(x) dx

show that to every M > 0 there is a c(M) such that

(2.3) I :~g(~ +i~)] <~c(M) (1 + I~ +i~71) -Meh('),

where h(~7) = maxxT, T6S(g)

is the so-called support function of S(g).

The estimate (2.3) leads naturally to a definition of an inverse Fourier-Laplace trans-

form by duality. Suppose that F(~+i~7) is hoIomorphie in R e Z + i A where A c R e Z is

open and connected and suppose that

(2.4) ]F(~+~T) I <c (T) ( l+ I~+iTI) N, TeA,

for some _AT and some C(T ) locally bounded on A. Then

(2 .5 ) /(x) = (2 ~)-nfF(~ + iT) e 'x(~+~') d~, T e A,

defines a distribution/, called the inverse Fourier-Laplace transform of F and denoted by

:~-1]. Here (2.5) has to be interpreted as

(/, ~) = (2 g)-nfF(~ + iT) ~g(~ + iT) d~, T 6 A, (2.5')

where ~(x)=g( -x ) and g6Co(ReZ' ). The estimates (2.3), (2.4) show that the integral is

absolutely convergent; by Cauehy's theorem, it is independent of T.

The construction of fundamental solutions of hyperbolic operators relies on the fol-

lowing well-known variant of the Paley-Wiener theorem.

2.5. THEOREM. Let F satis/y (2.4) and suppose in addition that A is convex and

(2.7) t A c A , C(tT) = C(T )

when t ~ 1 and let F = [.J sA /or s > O. Then the support o / / = :~-XF is contained in the cone K

dual to - I ~. I / A = F and (2.4) is replaced by

(2.4') I~(&+iT)l < c(T)(1 + I~+iTIV(1 + ITI-%

where now C(tT)=c(T ) /or all t>0 , then

126 M. ~. ATIYAH, R. BOTT AND L. G~RDING

(2.8) F_(~)=l imF(~§ r e 0 , ~ E - F

is a temperate distribution and

(2.9) / = :~-IF_.

1Vote. I t is easy to see tha t (2.4') completely characterizes Fourier-Laplace transforms

of temperate distributions with support in K = - F ' .

Proo/. We know the right side of (2.5') to be independent of ~ and we can majorize

the integrand using (2.3) and (2.4). I f S(~)rl K is empty and S(9 ~) is convex, there is an

~ E F such tha t x~>~0 on S(~), i.e. x ~ < 0 on S(g)= -S(~) and then (2.3) holds with no

exponential factor On the right. Moreover, t~ E A when t is large enough. Hence, replacing

by t~ in (2.5') and letting t--> 0% the integral vanishes. Hence S( / )c K = - F ' . When (2.4')

holds, then, fixing ~ E F, replacing ~ by t~ 7 and letting t r 0, our earlier remarks on distribu-

tions which are boundary values of analytic functions show tha t the limit (2.8) exists,

tha t it is a temperate distribution and tha t (/, ~) = (F, :~g). This proves (2.9).

Chapter 2. Fundamental solutions of hyperbolic operators with constant coefficients

Section 3 defines hyperbolic operators and states their main properties. In section 4,

we shall get fundamental solutions expressed as inverse Fourier-Laplace transforms. This

section ends with an outline of how to get the Herg lo tz -Pe t rovsky-Leray formulas. This

will be carried out in detail in section 7, which relies on a close analysis of the geometry

of hyperbolic surfaces including the semi-continuity of the local cones and a s tudy of the

wave front surface (section 5). The vector fields and cycles used in section 7 are presented

in section 6.

3. Hyperbolic differential operators and hyperbolic polynomials

Let X be an open part of R n and

P(x, D) = ~ P , ( x ) D ' , D = Dx = i-l~/~x,

a differential operator on X with smooth complex coefficients. A (right) fundamental solu-

tion of P is, by definition, a distribution E(P, x, y) on X • X such tha t

P(x, D~) E(P, x, y) = ~ (x -y ) .

P is said to be hyperbolic in X if E can be chosen so tha t i t vanishes when x is close to y


except when x belongs to some proper cone with its vertex at y. The same definition applies

to differential operators whose coefficients are square matrices of some fixed order.

We shall only deal with the case when P has constant coefficients. We say that P

is hyperbolic with respect to v~ = (V~l ..... v~n) ERe Z if P has a fundamental solution E=

E(P, ~, x) now satisfying PE=(~(x) and having support in some closed cone K with its

vertex at the origin such that xv ~ > 0 on/~ = K - (0}. Here the coefficients of P are supposed

to be r • r matrices and I is the unit r • r matrix. By a translation, this is equivalent to

hyperbolicity in the sense above.

3.1. LEM~A. When P is hyperbolic with respect to v~, the/undamental solution E(P, ~, x)

is unique.

Proo/. I t is well known that the space of distributions with support in a proper cone

K with its vertex at the origin forms a convolution algebra and that the space of distribu-

tions with support in a closed half-space whose interior contains/~, forms a module under

convolution with this algebra. (Schwartz 1950, II , p. 32.) The same holds for distributions

whose values are square matrices. Put P(x)=P(D)e)(x)I so that P(~/~x)/(x)= (D~/)(x).

Then P ~ E = 81 is the unit in the convolution algebra of such distributions with support

in K. Since the corresponding scalar convolution algebra is commutative, this shows that

also E ~ / 5 =~$I. Now let F be another fundamental solution of P with support in xO ~> 0.

Then F = ~ ~+ F = E~e_P~+ F = E. This finishes the proof.

The convolution algebra also gives a proof of the following lemma that reduces the

study of hyperbolic operators with constant coefficients to the scalar case.

3.2. LEMMA. A di//erential operator P(D) whose coe//icients are square matrices is

hyperbolic with respect to ~ i] and only i/ its determinant (det P)(D) has that property.

Proo/. Suppose first that P is hyperbolic with respect to v ~ and let E be the corre-

sponding fundamental solution. Then P ~ E = 8I and, taking determinants in the sense of

convolution, this gives det P~- det E = 8. Since S(det E) c K and det t5 = (det P)~, this

shows that det P is hyperbolic with respect to v% On the other hand, suppose this to be

true and let E(detP, v~, x) be the corresponding fundamental solution. Let Q(D) be the

matrix of cofactors in P(D) so that PQ = (det P)I. Then

(3.3) E(P, ~, x) = Q(D) E(det P, v~, x) I .

In fact, S(E) c K and

P(D) E(P, v~, x) = (det P) (D) E(det P, v~, x) I = ~(x) I .

128 i%I. F. ATIYAH, R. BOTT AND L. GARDING

For scalar operators P(D), hyperbolicity is equivalent to an algebraic property of its

characteristic polynomial P(~) = ~ P ~ "

of P. Assume P to be of degree m and let

be the characteristic polynomial of the principal par t Pm(a/~x) of P(~/~x).

3.4. Definition. The polynomial P(~) is said to be hyperbolic with respect to v q E Re Z

if Pm(vq)40 and P(~+t#)4=0 when ~ is real and I m t is less than some fixed number. The

space of such polynomials will be denoted by hyp (v q, m). When m is not specified we write

hyp (vq). A polynomial P(~) whose coefficients are r • r matrices is said to be hyperbolic

with respect to v q if det P(~) has tha t property. The space of such polynomials will be

denoted by hypr (vq).

We now have

3.5. TIZEORE~. A di//erential operator P(D) whose coe/ficients are r • matrices is

hyperbolic with respect to # i /and only i / i t s characteristic polynomial belongs to hypr(vq).

In view of Lemma 3.2, it suffices to consider the scalar case r = 1. The theorem is due

to Gs (1950). I t results from Theorem 5.6.1 and 5.6.2 in HSrmander 's book Linear

Part ial Differential Operators (1963), hereafter referred to as H. The book also gives short

proofs of the most important properties of hyperbolic polynomials. For the reader 's con-

venience, we shall repeat some of them.

We are going to sketch a proof of the necessity par t of Theorem 3.5. The sufficiency

will follow from the construction of fundamental solutions in the next section. We have to

show tha t if the operator P(D) is hyperbolic with respect to v ~ then P e h y p (v~). In fact,

let E(x) be a fundamental solution of P with support in a proper cone K such tha t xt$ > 0

on K. Let q eC~(Re Z') be 1 in a neighborhood of the origin. Then

P( D) /(x) = ~(x) + g(x),

where / = ~ E and g=P(D)@- 1)E are distributions with compact supports and

h(~)=maxx~, xE S(g)

is positive homogeneous in ~, h(2~)=2h(~) when +l/> O, and, since S(g) is a compact par t of

/~, h ( - v ~) <0. Talcing Fourier-Laplace transforms,

P(~) F(~) = 1 + G(~), ~e EZ,

where _~ = ~/, G = :~g are entire functions and

LACUNAS :FOR HYPERBOLIC D I F F E R E N T I A L OPERATORS

for some C, N > O. Hence

(3.6)

129

P(2) = 0 ~ h(Im 2) >~ - c log (2 + 121),

where c > 0 is another constant. In particular, P # 0 . Let Pm be the principal part of P.

If Pm(v ~) =0, it is easy to see that the equation P(2+tv ~) =0 has a root t=t(2) such that

Im t ~ - ~ as 2 tends to some 2 ~ along a suitable path. But then h(Im (~ + tvq))= h(Im tvq) +

0(1) tends to - ~ and this contradicts (3.6) with 2 replaced by ~ +tv q. Hence P~(vq)#0.

Further, (3.6) implies

P(~ +tv~)=O:~h(Im tv~) >~ - c 1 log (2+ I~1),

where now ~ is real and c I is a new constant. A well-known lemma by Seidenberg-Tarski

(H Appendix, Lemma 2.1) shows that the logarithm in the right can be replaced by a

constant and this finishes the proof.

We shall mainly be concerned with homogeneous hyperbolic polynomials a(~). In view

of the definition above and the homogeneity of a, we have a E hyp (v~, m) if and only if

(3.7) 2 real, I m t ~0 ~a(2 +tvq) #0

or, equivalently,

(3.7') ~ real =~a(~+t~) = 0 has m real roots t.

3.8. Definition. The space of homogeneous polynomials a e h y p (v ~, m) will be denoted

by Hyp (v~, m). An a e H y p (v~, m) is said to be strongly hyperbolic if the zeros of a(~+t~)

are all different when ~ is not proportional to v q. The space of these polynomials will be

denoted by Hyp ~ (v ~, m).

When m is not specified, we write Hyp (v q) and Hyp ~ (vq). Note that Hyp (v~)=

Hyp ( - ~ ) .

The interest of strongly (or strictly) hyperbolic polynomials is that they remain hyper-

bolic under addition of arbitrary lower order terms, i.e.

(3.9) a f H y p ~ (v ~, m), degree b<m ~ a + b e h y p (v~, m).

This will be proved later. We shall sometimes let hyp ~ (v ~, m) denote the space of poly-

nomials in hyp (vq, m) whose principal parts are strongly hyperbolic. This class also gives

rise to hyperbolic operators with variable coefficients. In fact, if X c R ~ is open and

P(x, ~]ax) is a differential operator with smooth coefficients and P(x, ~)fhyp ~ (v ~, m) for all

x fiX, then, locally in X, P has a fundamental solution with the properties required by the

general definition of hyperbolicity (H Theorem 9.3.2). 9 - 702909 Acta mathemat~ca. 124. Imprim6 Io 13 Avrll 1970.

130 ~ . F. ATIYAH, R. BOTT AND L. G~RDING

Homogeneous hyperbolic polynomials. I t follows from (3.7') that if aEHyp (O), then

a(~)/a(O) has real coefficients so that a restriction to real polynomials is not essential.

All polynomials in the following examples are supposed to be real.

3.10. Examples. The class Hyp (O, 0) consists of all non-vanishing constants. A linear

polynomial bl~+.. , bn~ n belongs to Hyp (O, 1) if and only if b040 . The quadratic poly-

nomial a ( ~ ) = ~ - b 2 ~ - . . . - b j ~ belongs to Hyp (O, 2) with O=(1, 0, ..., 0) if and only if

b~ ..... b~ ~> 0 and it is strongly hyperbolic if and only if b 2 ..... bn > 0. More generally, it is

easy to see that a quadratic polynomial a(~) belongs to Hyp (O, 2) if and only if a(~)/a(O)

has Lorentz signature 1, - 1 , - 1 , ..., 0 .. . . and O is a time-like vector for a(~)/a(O). I t is

important to note that Hyp (O) is closed under multiplication,

(3.11) al, a 2 E H y p (O) =~ala2EHy p (O),

and that factors are hyperbolic,

(3.12) Hyp (0) ~ a = ala 2 . . . =~al, a 2 .... EHyp (0).

Note that (3.11) fails in general for Hyp ~ (0). If, e.g. aEHyp ~ (0), then a2EHyp (0) but

is not strongly hyperbolic. On the other hand, factors of a strongly hyperbolic polynomial

are strongly hyperbolic. Hyperbolic polynomials can also be obtained by polarization.

If a E Hyp (0, m) and m

a(~ § tO) = ~ t~ak (~) 0

then ak EHyp (O, m-Ic), O <-~b <-~m. In fact, ak(O)=(mk) a(O) 4O and, since the polynomial

t-~a(~ +tO) has only real zeros, the same holds for all its derivatives. Since, e.g., ~15~ ... ~n E

Hyp (0, n), 0 = (1, 1 . . . . . 1), it follows from this that the elementary symmetric sum

~1 ~2..- ~k belongs to Hyp (0, k). Localizations of hyperbolic polynomials are hyperbolic

(see Lemma 3.42).

I t is obvious that if a(~) is real and homogeneous and strongly hyperbolic with respect

to 0 so is b(~) if b is real and homogeneous and sufficiently close to a and degree b =degree a.

More generally, we have the following result (W. Nuij 1969) where, temporarily, Hyp (0,m)

and Hyp ~ (0, m) are restricted to real polynomials.

3.13. Lv.MMA. Let H (m) be the space o/non-zero real homogeneous polynomials o/degree

m. Then Hyp ~ (O, m) is open in H(m) and Hyp (O, m) is the part o/the closure o/Hyp ~ (0, m)

where a(O) 40. Both Hyp (O, m) and Hyp ~ (O, m) consist o/two connected and simply con.

nected pieces determined by the sign o/a(O).

When a(~) is a homogeneous polynomial, let A be the complex hypersurface a(~)= 0,


~ E Z = C ~ and Re A its real par~. The surface A is said to be hyperbolic when a E H y p (v ~) for

some v ~. The figure 3 shows the image Re A* in real projective space of a typical Re A.

The picture also shows Re B* when b is strongly hyperbolic and close to a.

The cone F(A, v~).

3.14. Definition. When a E Hyp (v~), let F = F(A, v~) be tha t component of Re Z - R e A

which contains v ~ (see figure 3).

By the homogeneity of a, F(A, -v~)= - F ( A , v~). I t is clear tha t F is an open cone,

tha t F = Re Z when a is a constant and tha t

(3.15) F(A, 0) = f l F(A~, 0)

when a = a 1 a S ... is a product.

When a(~) ~=0, let us factorize

(3.16) a(~ § = a(~])1~ (t +,~k(~, ~)).

Then a is hyperbolic with respect to 0 if and only if all 2k(0, $) are real when $ is real and

strongly hyperbolic if and only if these numbers are real and different when $ is not pro-

portional to ~. Further, a real $ belongs to F(A, ~) if and only if all 2k(O, $) are positive.

/

,'/ % ~ d I I S

"\

1 I

�9 'ig. 3. Pic ture of Re A* , a E H y p (v ~, 7), n = 3 . E v e r y s t ra igh t line t h rough ~* mee t s Re A * in seven real points. The dotted lines indicate Re B* when b is strongly hyperbolic and close to a. r* is the image in real projective space of r(A, ~).

Algebraic properties o/ hyperbolic polynomials. I t is obvious tha t H y p ( - v ~, m ) =

H y p (v~, m). The following result is less obvious but easy to prove.

3.17. L ~ A . hyp ( -v~, m ) = h y p (v ~, m).

Proo/. Let P E h y p (v a, m) and let a =Pro be the principal par t of P. Then a(v~)4 0 and

there is an s o such tha t

132

(3.18)

Hence, if we factorize

(3.19)

(3.18) means that

(3.18')

M. F . A T I Y A H s R . B O T T ALND L . G/~RDING

~e reaI, Im s <So* P(~+sv q) 4= O.

P(~ + sO) = a(O) 1-I (s +/~k (0, ~)),

rea l*Im/~k(0, ~)~> -s0, Vk.

Since a(-z$)=(-1)ma(~9)4:0, it suffices to show that there is an upper bound to the

imaginary parts in (3.18'). Now/~(~) =~/~k(0, 2) and hence also Im/~(~) is a polynomial in

of degree one. According to (3.18'), Im/,(~) is a constant c and also a sum of m terms which

are bounded from below by - s 0. But then no term exceeds c+ ( m - 1)s 0 and this finishes

the proof.

3.20. L~.~MA. I / P belongs to hyp (v a, m), its principal part a belongs to Hyp (~9, m).

Proo/. We have, with ~ a n d , real,

a($ +t~9) = lim z-mP(z~ + rtO), ~ ~ + co,

where all the imaginary parts of the zeros of the polynomials to the right are bounded by

c~ -1 where c is a constant. Hence the polynomial t-->a(~+tv (~) has only real zeros and, since

a(v~) ~:0, there are m of them.

The preceding lemma motivates

3.21. De/intion. When P e h y p (v~), put F(P, ~9) =F(A, ~9) where a is the principal part

of P.

Note that F(P,/~) and F(P, - ~ ) are opposite cones. The following simple lemma is

basic (Gs 1950, H Theorem 5.5.4).

3.22. Lv.MMA. Let P E h yp (~9, m) so that (3.18) holds and let ~ E F (P, ~ ). Then

(3.23) real, Im t < O, Im s < S o ~ P(~+t~ +sO) 4: O.

Proo/. Since (3.18) holds, the polynomials

(3.24) t -~u-mP(~+tu~+(s+i(1 -u))v~), Im s<so, ~ real, u~>l

have no real zeros. In particular, they have a constant number of zeros in the lower half-

plane. To compute this number note that , as u-~ co, the polynomials (3.24) tend to

LACUNAS FOB HYPERBOLIC DIFFERENTIAL OPERATORS 133

t->a(t~ -iO) = a(O) 1-I( - i + t2~(v ~, U)),

where a is the principal part of P and (3.16) is used. Now since ~EF(A, #), all 2k(v~, 7) are

positive so that all the zeros of this last polynomial lie in the upper half-plane. Hence all

polynomials (3.24) share that property and, putting u = l , we have the assertion of the

lemma.

3.23. C o R O L L AS Y. I / P E hyp (O) and ~7 E F(P , v~), then P E hyp (7). The cone F(P, v ~) is

c o n v e x .

Proo/. If ~ E F(P, v~), then ~ -eO E F(P, O) if e > 0 is small enough. Hence, by Lemm a 3.22,

P(~ + tr/) = P(~ + t(r/- eO) + tev a) 4:0

when Im t < m i n (-s0/e, 0). Also, specializing Lemma 3.22 to the principal part a of P, we

have s~>O, t~>O, s + t > O ~ a(~+it~+isO) 4=0,

when ~ E F(A, O) = F(P, 0). Since a E Hyp (~), for any ~ E F(A, ~), we may here replace v~

by such a $ and this shows that F(P, }) is convex.

When P E hyp (0), the fundamental solution E(P, v~, x) will be constructed as the inverse

Fourier-Laplace transform of P(} + i~) -I where U belongs to a certain subset of - F ( A , v~).

The following corollary contains all the necessary information for the construction of

such fundamental solutions.

3.24. COROLLARY. Let PEhyp (v~). Then there is an Sl>O such that every/unction

(3.25) ~+in->P(~+i~+isO) -x, e(Re s -es l ) >0, e= •

is holomorphic and bounded when ~ +i~ERe Z +eiF(P, 0). When P = a is homogeneous, the

/unctions +i~ -~ a(~ + iv)-~

are holomorphic in Re Z • ~) and satis/y the inequality

(3.26) l a(, +i )-11 < l a(v)]-l.

Proo/. The functions (3.25) are of course holomorphie where P does not vanish. Re-

placing ~, s by ~+i~, is in (3.19) gives

P(~ +i~ +~a) = aO) YI (~ +~(0, ~ +i~)).

By Lemma 3.22, Im pk(v~, ~ + i~) is bounded from below when ~ is real and ~ E -- F = - F(P, v~).

134 M. F . A T I Y A H , R . BOTT A N D L. G/~RDING

Hence there is an s o such that the functions (3.25) are bounded when ~ + i ~ T E R e Z - i F

and Re s <s 0. Changing v~ to - v ~ and noting that, by Lemma 3.17, P E h y p ( - v ~) and that

F(P, -zT)= - F ( P , v~), the first part of the corollary follows. The second part is obvious.

To understand the role of this corollary, the reader should now study Theorem 4.1

before proceeding further. The fundamental solution E(P, z$, x) constructed there has its

support in the cone K = K(P, z?)~ Re Z' dual to F(P, zg).

Puiseux series. Some of the deeper properties of hyperbolic polynomials result from

expanding zeros in Puiseux series. Our next lemma collects material of this kind.

3.27. L ~ M A . Let P E h y p (0, m) and let aEHyp (z~, m) be the principal part o /P . Let

~, ~ E Re Z. Then the polynomials

s, t-+ a(~ +t~ + s~)

s, t --~ P(~ +t~ +sO)

can be [actorized in the [orm

m

(3.28) a(~ + t~? + svg) = a(~) 1-I (s + 2k (vg, ~ + t~)) 1

(3.29)

where the ]unctions

(3.30)

m P(~ § tr/§ sv~) = a(v~) 1-I (s +/z k (v~, ~ + tr/)),

1

R 9t--~2k(zg, ~+t~)

are real and analytic with simple poles at oo so that

(3.31) 2~(~, ~+t~) = t~k(o, ~ )+o(1 ) , t - + ~ .

When ~ E I ~ =F(A, v~), their derivatives are positive, when ~ E F they are non-negative. The

[unctions

(3.32) R 9 t --~/tk(v ~, ~ +t~)

can be labelled so that, ]or large t,

(3.33) ttk(~, ~ +t~7) = t2k(v~, ~) +0(1).

Proo]. Close to any real to, the functions (3.30) can be developed into convergent Pui-

s e u x s e r i e s

(3.34) ~k(~, ~ § =2k(~, ~ +to~7) +ck(t--tO)'k (1 §

LACUNAS ~OR HYPERBOLIC DIFFERENTIAL OPERATORS 135

where ck ~:0 and r k > 0 is rational. All the branches of these series being real when t is real

implies tha t rk is an integer and also tha t the series are power series in t - t o with real

coefficients. Varying t o and making an analytic continuation proves the first s ta tement of

the lemma. Taking t large and noting tha t

t---> c~ ~t-mP(~ + t~ I +tsv~)~a(~ +sO),

where the convergence refers to polynomials in s, it follows tha t

(3.35) #k(O, ~ +t~) = t2k(v~, 7) +o(t), 1 <<-k <<.m

if the # ' s are labelled suitably. Now the left sides can be developed in Puiseux series in

descending powers of t 1/m starting with t. Since by the hyperbolicity of P, their imaginary

parts are bounded, no series can contain a positive power of t 1/m except the first term of

(3.35) and hence (3.31) follows. When this reasoning is applied to the real functions (3.32),

(3.33) follows. Finally, if ~/EF=F(A,v~), then, since r is convex, I m (t~+sv q) belongs to

F when I m t~>0, I m s>~0, I m ( t+s)>0 and hence, by Lemma 3.22,

I m t > 0 ~ I m 2k(O, ~+t~) > 0.

This is possible only if rk = 1 and c~ > 0 in (3.34) and hence the functions (3.30) have positive

derivatives. A slight modification of this argument shows the derivatives are /> 0 when

Multiplicities and localization

3.36. De/inition. Let P(~) be a polynomial of degree m>~0 and develop tmP(t-l~+~)

in ascending powers of t,

(3.37) traP($-l~ + ~) = t~P i(r + O(t 1~+I)

where Pg(~) is the first coefficient tha t does not vanish identically in $. The number p = mE(P )

is called the multiplicity of } relative to P, the polynomial ~-~Pg(~), the localization of

P at }.

Note tha t if P =P0 +..- +Pro is a decomposition of P into homogeneous parts, Pk(2}) =

;tePk(}), then (3.37) reads

(3.38) ~. tm-kPk(~ +t~) = trP~(r O(tY+l) .

In particular, when P = a is homogeneous,

(3.39) a(~ + t~) = t~ad~ ) + O(t~+~).

I t is obvious tha t

136

(3.40)

(3.41)

M. F. ATIYAH~ R. BOTT AND L. G~RDII~G

m~(PQ) = mi(P) § m~(Q)

(PQ)~(~) = P~(~)Q~(~)

when P and Q are (non-vanishing) polynomials.

The concept of localization extends to non-zero rational functions / ~ Q/P,

(3.37') t,n/(t-1 ~ § ~) = t~/~ (~) + O(t~ +1).

Here the right side is a formal power series in t whose coefficients are rational functions

of ~, the localization /~ is the first non-vanishing coefficient, m=m(/)=re(Q)-m(P) is

the degree of / and p = m~ (/) is the multiplicity of / at ~. One has )r = QdP~ and m E (/) =

m~(Q) -m~(P).

Examples. Let a be the principal part of P. If ~=0, then P~(~) =P(~). If a(~) ~:0, then

P~(~) =a(~) is a non-zero constant. If a(~) =O but grad a(~) ~=0, then

P~(~) = grad a(~) ~ + const

is a polynomial of degree 1. When P is strongly hyperbolic, these examples exhaust the

possibilities for P f with ~ =~O. When P is not strongly hyperbolic, the degree of such a

s may of course exceed 1. If P = a is homogeneous, then a[(~)=0 defines the tangent cone

A~ of A: a(~)--0 at ~.

Localizations of non-homogeneons polynomials in the sense given above have been

used by HSrmander (1969) in a study of singularities of fundamental solutions for arbitrary

differential operators.

Our next lemma shows that localizations at real points of hyperbolic polynomials are

hyperbolic.

3.42. L v, ~ M A. Let P E hyp (~) with principal part a and let ~ E Re Z. Then

(3.43) m~(P) = mE(a )

(3.44) a~ is the principal part o~ PE

(3.45) P~Ehyp (0), a~EHyp (0)

(3.46) A ~ depends only on the double ray ~

(3.47) F(AE, ~)D F(A, ~).

Proo[. Let us first consider a. Putting s = 0 in (3.28) and using the differentiability in

(3.30) we get


a(~ + trl) = a(v q) ~I (]~k(v a, ~) + t% + O(t2)), 1

where t is small and all c~>0 when ~/EF=F(A, vq). Since F is open, a comparison with

(3.39) shows tha t

(3.48) ~EF ~a~(~) # 0

and that precisely p =m~(a) of the numbers 2~(v~, ~) vanish. Now, by the preceding lemma,

t" P( t-~ ~ +*1) = a(O) 1-I ttt~ ( t-~ ~ + 7) = a(O ) [-I ( ~ (O, ~) -4- O(t) )

when , / is real. Hence tmP(t-~ +*1) = O(t€

and this shows that m~(P)>~p. Hence, developing the left side of (3.37) in a Taylor series

around t-i t , we get m

t~P(t-i~ + 7) = 5 ~/~(~), P

where the/k are polynomials in U of degree ~< k depending on ~ as a parameter. Replacing

t by ts -1 and ~ by s~ and letting s-~ oo this gives

rn

t~a( t-1 ~ + 7) = Y ~/(~) (~), P

where /~k)(~) =l im s-k/~(s~), s ~ oo,

is the homogeneous part of/~ of degree k. A comparison with (3.37) now shows that ](~') (7) =

a~(~). Hence/~ (7) does not vanish identically and this means that P~ (7) =/~(~) has the princi-

pal part at(~). This proves (3.43) and (3.44). By the definition of P f and p = m i ( P ) ,

(3.50) R ~ t ~O :~ t " - ' P ( t - ~ + 7 +sO) = P ~(~ +sO).

Since PEhyp (0), there is an s o such that the polynomials

s -~ t"-~'P(t-z~j +r/+sO)

have no zeros s with I m s < s 0 when ~ is real. According to (3.50), the polynomials s-~

P~(~ +sO) have the same property. Since (3.44) holds and, by (3.48), a~(O) #0, this proves

(3.45). Finally, (3.47) is a consequence of (3.48) and the definition of F(A~, O) and (3.46) is

a simple consequence of (3.39) and the homogeneity of a, one has

a;t~(~/) =).m-Va~(~) when 0 #)~ E R.

In passing we shall now prove a quantitative result, to be used later.

3.51. L~.~MA. Let P e h y p (O), p=m~(P) , ~EReZ and choose s o such that P(~/ + ~ ) # 0

138 M . F . ATIYAH, R. BOTT AND L. Gt~RDING

when ] Im s ] > s o. Then there exist positive numbers c and 2V depending only on s and ~ such

that, i /~ is real,

0~<t~< 1, I Im sl>So~lt'~-vP(t-l~+~+sv~)l >c(1 + l r -N

t ~ o ~ tm- 'p( t - l ~ + r + sa) ~ Pe(~ + s#).

Proo[. Fixing s and ~, let us put

R(t, r = Itm-'e(t-X~ +r +S~9)12.

Then R is a polynomial in ~ and ~/and, since R(0, r = tPe(r then

~(r) = in~ R(t, r < ir~ R(t, r

is positive for all v. Now by the well-known Seidenberg-Tarski theorem (see Lemma 2.1

of the Appendix in H), for large v, ~0(r) is not less than a positive constant times some

negative power ~. This proves the lemma.

Lineality and reduced dimension. A hyperbolic polynomial P(~), ~EZ, may depend on

less than dim Z variables. The lineality L(P) of P in Z is, by definition, the space of ~ EZ

such tha t P(t~+r162 identically in r t or, equivalently, P~(r162 for all r i.e.

me(P) = degree P. I t is clear tha t L(P) is a linear space and tha t

L(PQ) = L(P) N L(Q)

when P, Q are not identically zero. Hence, writing P as a product of irreducible factors,

we see tha t if P=~0, then L(P) depends only on the complex surface P(~)=0. When P = a

is homogeneous we shall therefore write L(A)=L(a) where now A is the complex surface

a(~) = 0. I t is obvious that P is a polynomial P" on the quotient Z/L(P) and tha t the lineality

of P ' in the quotient vanishes. The dimension of the quotient, i.e. the codimension of L(P)

in Z, will be called the reduced dimension of P and denoted by n(P). When P = a is homo-

geneous we also write n(A)=n(a). A polynomial whose lineality vanishes is said to be

complete. We have

3.52. LEMMA. Let P E h y p (~) and let a be the principal part o /P . Then L(P)=L(A),

n(P) =n(A) and Re L(A) equals the edge o / F = F(A, 0), i.e. Re L(A) consists o/ all ~ ERe Z

such that

(3.53) F + R~: = F.

1] a~ is the localization o] a at ~EZ, then ~EL(A~).


Proo/. By Lemma 3.42, mE(P)=m~(a ) when ~ is real. In particular, R e L ( P ) =

Re L(A) so that, clearly, L(P)=L(A). Let us factorize

a(s U + ~' § t~) = a(~) l~ (s + ~k(7, ~' -t- t~) ), ~ E F. 1

By the definition of F, ~ satisfies (3.53) if and only if

(3.54) 2k(U, U' + t~) >0, u U' E F, Yt E R.

If ~ERe L(A), then 2k(~, 7' +t~) =~k(~, ~') >0 and the condition holds. Conversely, if

(3.54) holds, dividing by t and letting t - ~ , we get ~k(~, ~)=0 for all ~EF so that also

a(u + t~) = a(u) for all t and ~ E F. But then, since F is open, ~ E Re L(A). By the definition

of a~, a(~ +t($ ~- s~)) = af($ -]- s~) t ~ ~- O(t~+l),

where p = m~ (a). But the left side also equals

(1 + ts)'na(~ + (1 + ts)-lt~) = a~(~)t'(1 + ts) m-~ + O(t p+I)

Identifying the coefficients of t ~ of the right sides proves the last statement of the lemma.

The propagation cone. When PEhyp (v~), we know that the convex hull of the support

of the fundamental solution E(P, ~, �9 ) is the cone dual to F(P, v ~) = F(A, ~) where a is the

principal part of P. This cone will be called the propagation cone, its formal definition

being

(3.55) K(P, ~) = K(A, v~) = (x; xERe Z', xF(A, ~) >~ 0).

We state some of its properties

3.56. Lv.~MA. Let aEHyp (v% Then

a) K = K ( A , ~) is a proper closed convex cone with its vertex at the origin and

r = F(A, ~) = {~; ~ e Re Z, ~$: > 0}, $: = K - {0}.

b) K is contained in and spans the orthogonal complement

L~ = (x; xERe Z', xL(A)=0}

o/the lineality o/a. Hence dim K=n(A) .

c) The interior o / K relative to L~ consists of all

(3.57) Va(~), ~EF

140 M. 1~. ATIYAIt, R. BOTT ~ND L. G/~RDING

and also o] all

(3.58) VV~ log a(~) -x, ~ q F

Here V =grad, Vr = (7, grad), ~ is any vector in F and a is normalized so that a(O) >0.

Proo/. a) and b) follow from the preceding lemma and our earlier remarks about the

duality of cones. In view of b), it suffices to prove c) when a is complete. We know then that

if x E/~, the intersection of F with any half-space x~ <c is compact. In particular, if B is a

closed part of F, the infimum c of x~ on B is positive and the hyperplane x~=c touches B

at some point ~ =~(x)fiB. Taking B to be the hypersurface a(~)= 1, ~ F and noting that,

since a has non-zero homogeneity, Va(~) does not vanish in F, we see that every x in

is proportional to some Va(~), ~7 fiB. Clearly, the proportionality constant is positive and

this proves that every x in k has the form (3.57). Actually, B is strictly convex (see G~rding

(1959)) so that the parametrization is unique. Similarly, letting B be the hypersurface

F(~) = V~ log a(~) -~ = - 1, $ q F, it follows that every x i n / ~ has the form (3.58). In fact,

factorizing

(3.59)

all Xk are positive and

(3.60)

a(~ + t~) = a(~) 1~ (t + ~k (7, ~)), 1

/n

~(~1 = - Y ~ ( 7 , ~1-1 1

so that B is not empty. Since F has homogeneity - 1, VF 40 in F. I t remains to show that

all vectors (3.57) and (3.58) are in/~. Writing (3.59) in the form

a(~ +t~) -= a($) 1-[ (1 +t2k(~, ~/))

we have, if ~EF and ~ E R e Z

V~a(~) = a(~) (~1(~, 7) + ... +~m(~, 7))"

Now, if 0 #~ E P, all 2k are non-negative and at least one is positive. This shows that the

linear form ~-->V~a(~) is positive on F - { 0 } and hence Va(~)Ek. Next, let 7, ~EI" and

~EReZ. Replacing ~ in (3.60) by ~+t~ and taking the derivative with respect to t at t=0 ,

we have v ~ ( ~ ) = Y~ ~(~, ~)-~c~(~, ~, ~),

where 2k(~, ~ +t~) =2k(~, ~) +tck(~l, ~, ~) + O(t~).

By Lemma 3.27, all c~ are >~ 0 when ~ E F and hence also V~ F($)/> 0. If all ck vanish, then


m m

a(~ + t~) = a(~) ~ ~ (~, ~ + t~) = a(u) YI (~ (~, ~) + O(t~))

so that VCa(~)=0. But then, by a previous argument, ~ has to vanish. Hence the linear

form ~-~VCF(~) is positive on ~ - { 0 } and this shows that all gradients (3.58) are in /~ .

The proof is finished.

Local cones. The wave/ront sur/ace. We shall introduce the cones and linealities associ-

ated with the localizations of a hyperbolic polynomial.

3.61. Definition. w h e n P Ehyp (v~), ~eRe Z, put

Fg(P, v~) = F(P~, v~), K~(P, ~) = K(P~, z~), L~(P) = L(P$), ng(P) = codim L$(P).

Examples. Let a be the principal part of P. When ~ = 0 we have the uninteresting case

P~ =P. When a(~) #0, then Pg(~) =a(~) is a constant and I~ = R e Z, K~ = {0}, n~ =0. When

a(~)=0 but Va(~)#0, then a~(~)=V~a(~)=~a(~) /~k so that I~ is the half-space

a~(z$)-la~(~) >0, n~ = 1 and K~ is the half-ray spanned by a~(v~)-lVa(~). When P is strongly

hyperbolic and ~ #0, these examples are exhaustive. When P is not strongly hyperbolic,

F~ may be smaller than a half-space and K~ larger than a half.ray and n~ > 1. Note that

F~=F , K~=K when ~EL(P).

By Lemma 3.42, F~(P,v~)DF(P, vq) and hence also KE=K(P,~ ) for all real ~. The

local propagation cones K~ generate the wave front surface

W = W(P, vq) = W(A, ~) = [.J K~(P, 0), ~ #0,

which will be dealt with in detail later. I ts role as the carrier of the singularities of the

fundamental solution E = E(P, ~, x) is explained in the introduction. In particular, it

will turn out tha t the restriction of E to K - W is holomorphic. When L(P)#0, then

W = K so that W is not an interesting object in tha t case. When L(P) = 0, i.e. P is a complete

polynomial, then, as we shall see later, W is a closed subset of K containing the boundary

of K and contained in an algebraic hypersurface. The same would be true in general with a

different definition of W, viz. as (J K~(P, v~) for ~-EL(P).

Hyperbolic polynomials with a given principal part. Suppose that a E H y p (vq, m) is

strongly hyperbolic and that the degree of b is less than m. Then

(3.62) sup~l(ba-X)($+sz~)[ = O ( [ I m s]-l) , ~ real, Im s-~r

In fact, it suffices to prove this when ~ is restricted to some complement Y of Rv ~ in Re Z.

By partial fractions,


(ba -1) (~ +sO) = a(O) -1 ~ Bk($) (s +~k) -1,

where the 2k=2k(O, ~) as defined by (3.16) are real and

Bk(~) =b(~ - ]~kO)/ I-i (]tg -,2~). 1r

Since a is strongly hyperbolic, 2k(O, ~) 4~j(O, ~) when ] =t=/c and 0 =t=~ E Y and an easy argu-

ment shows that the Bk are bounded for large ~ so that (3.62) follows.

I t is clear that (3.62) implies that (a+b)(~+sO)~0 when ~ is real and Im s is large

enough and hence a +b E hyp (O). In the general case, (3.62) characterizes every b of degree

less than a such that a +b E hyp (O). We have the following result due to Leif Svensson

(1970).

3.63. THEOREM. I/ aEt Iyp (O) and the degree o /b is less than the degree o /a , then

a + b E hyp (O) i] and only i/(3.62) holds.

Later we shall use this theorem, but the proof is rather long and will not be reproduced

here.

4. Fundamental solutions of scalar hyperbolic operators. Localization.

Analytic continuation

We shall first construct fundamental solutions of scalar hyperbolic operators amd state

their simplest properties.

4.1. THEOREM. Let P E h y p (O). Then

(4.2) E(P, x) = E(P, O, ~) = (2 ~)-n f a e z P(~ + i~)-1 e~(e+f') d~,

where ~E - s O - F ( P , O) with s large enough, is the/undamental solution o / P with support

in K = K(P, O). No smaller convex cone than K contains the support o /E(P , x). When P = a

is homogeneous, then also

(4.3)

where

(4.4) a_(~) -1 = lira a(~ +it~) -1, t ~ O,

When P = a + b where a is the principal part o /P, then

E(a, 0, x) = (2 :~)- nfR e z a_ (~)-1 etX~d~,

e - F(A, 0).

LACUI~AS ~ OR H Y P E R B O L I C D I F F E R E N T I A L OPERATORS 143

(4.5) E(P, ~, x) = ~ ( - 1)kb(D)~E(a k+l, ~, x), 0

where the series converges in the distribution sense.

I / t > O, then

(4.6) E(a, v ~, tx) = tm-n E(a, v ~, x)

while, i/ t ~ O,

(4.7) tn-mE(P, ~, tx) -~ E(a, O, x)

in the distribution sense. I / the coordinates x 1 . . . . are chosen so that the/irst q = n(P) axes

span the orthogonal complement o/L(P), then

(4.8) E(P, ~, x) = E(P', ~', x')~(x"),

where x '= @1 ..... xq), x"= (xq+l .. . . ), P' =P(D').

Note. P'(~') is the restriction of P(~) to a complement of L(P) and it is a complete

polynomial. The formula (4.8) shows that it suffices to compute fundamental solutions

for complete hyperbolic operators in order to have them for all hyperbolic operators.

Proo/. Note that (4.2) has to be understood as the corresponding equality when both

sides are applied to a test function g E C o (Re Z),

(4.2') (E, ~) = (2 ~)-~ fae z P(~ § i~)-1 ~g(~ A- i~) d~.

With this understanding, the first part of the theorem is a consequence of Theorem 2.6

and Corollary 3.24. The same applies to (4.3). In fact, we only have to verify that E is a

fundamental solution, but this follows from (4.2') if we replace E by P(D) E. Then, by (2.2),

(P(D) E, ~) = (E, (P(D)g) v) = (2~)-~f:~g(~ +i~)d~ = g(0).

Next, let K o c K be a closed convex cone containing the support of E(P, x) and choose an

ERe Z such that ~x >0 on/~0. Then, by the first part of Theorem 3.5, P Ehyp (~) so that ,

by Lemma 3.20, a E H y p (~) where a is the principal part of P. I t follows that a (~)#0

when ~ belongs to the interior F 0 of the dual cone of K o. Now, since K 0 c K, we have

Fo~ F = F ( P , ~). But a(~) vanishes on the boundary of F and hence F o = K so that Ko=K.

To prove (4.5) note that, formally, (4.5) results from (4.2') by an expansion in a geometric

series

144 M . P . AT/YAH, 1~. BOTT AND L. G,~I~DI~TG

p-1 ($ + i~/) = ~ ( - 1)~b($ + i~) k a($ + i~) -~-1. 0

The convergence of this series when ~ =sv q with s large negative results from Theorem 3.63.

Moreover, the convergence is uniform in ~. Hence (4.5) follows. (4.6) is an obvious con-

sequence of (4.2) when P = a is homogeneous of degree m. To prove (4.7) observe that

tn-'nE(P, v q, tx) is the fundamental solution of Pt(D)=tmP(t-lD) and that P~(D)=a(D)+

b~(D) where bt(D ) =t'nb(t-lD). By Theorem 3.60,

tmb(t-l~-t-lis~9)a(~--isO) = b(t-l~-t-lis~9)a(t-l~-ist-l~9) = O(Re st -1)

when st -1 is large. Hence, fixing s and letting t $ 0, the series (4.5) for Pt converges in the

distribution sense to its first term. Finally, (4.8) is an immediate consequence of (4.2').

Localization o//undamental solutions. When P E hyp (zg) and P~ E hyp (v~) is the localiza-

tion of P at ~ E Re Z, consider the corresponding fundamental solution

(4.9) E~ = E~(P, ~9, x) = E(P~, ~9, x).

According to what we have just proved,

S(E~) ~ K~ = K(P~, ~9)c K = K(P, ~).

Let a be the principal part of P. As remarked before, K~ = {0} when a(~) 4:0 and, when

a(~) =0, grad a~ 4:0, then P~ is a polynomial of degree 1 and K~ is just the positive half of

the normal (grad a(~).v~) -1 grad a(~) to A: a(~)= 0, at ~. The following theorem, where SS

denotes 'the singular support of' shows that in a sense E~ is a localization of E = E(P, ~9, x).

The idea of localizing fundamental solutions by taking limits like (4.11) is due to Lars

HSrmander; our next theorem is just a special case of a general theory (HSrmander (1969)).

4.10. LOO~T.IZ~TIOI~ TttEOR~M. Let P ~ h y p (0, m), let E, ~, Es be as above and let

p =m~(P) be the multiplicity o/~ relative to P. Then

(4.11)

in the distribution sense and

(4.12) . 0 ~ S(E~) c SS(E).

Proo]. Letting ~ be a new variable of integration in (4.2) we have

E~ (x) = e-"X~E(P, ~) = (2 ~)-" f~, z P(t~ + ~ + i~)-1 e,X(~ +,,) dE.

Hence, by (4.2'),

LACUNAS FOR H Y P E R B O L I C D I F F E R E N T I A L OPERATORS 145

(tin-PEt, ~) = (2 ~)-~ fa~ z (t€ mP(t~ + ~ + i~))-~g(~ + iT) d~.

Putt ing ~] = - s v ~ with s large negative, Lemma 3.51 shows tha t the right side tends to

f P,(r +in)-l +in)dr (E,,

This proves (4.11). Next, let V be the complement of SS(E). If gECo(V ) and ~ 4 0 , then,

obviously,

f tm-~ e-~t~ E(P, x) g(x) dx

tends to zero as t-~ co. Hence S(E~) is contained in the complement of V and this finishes

the proof.

I t follows from (4.10) tha t

SS(E) ~ 19 S(E~).

So far we have been unable to find a case where this inclusion is proper.

The localization theorem extends to derivatives of the fundamental solution. The

precise s ta tement is as follows, the proof is the same as before.

4.10'. LOCALIZATION THEORV, M. Let E ( P ) = E ( P , zg, x) be the /undamental solu-

tion o / P E hyp (vg) let Q be a polynomial and put F = Q(D) E, F~ = Q~(D) E(P~), ~ real. Let

m( / )=m(Q) -m(P) be the degree o / / = Q / P and m~(/)-~m~(Q)-m~(P) the multiplicity o/

/ relative to ~. Then

(4.11') R 9 t ~ ~ ~ tm~(1)-m(r) e-ttx~ F(x) -~ F i(x)

in the distribution sense and

(4.12') ~ r 0 ~ S(F~) c SS(F).

The homogeneous case. Analytic continuation. Our further analysis of fundamental

solutions rests upon certain explicit formulas which show them to be holomorphic outside

the wave front surface (Theorem 7.5). In order to prepare for the proofs, it is conven-

ient to interpolate between all E(a ~, v~, x), (k = 1, 2 . . . . ), using a complex parameter s.

We put

(4.13) E8 (a, x) = (2 ~)-n f a(} § i~) -s e 'x(f+'',d}, ~ e - F(A, •),

10 -- 702909 Acta mathematica. 124. I m p r i m 6 le 8 Avr i l 1970.

146 • . 1~. A T I Y A H , R . BOTT A N D L. G~kRDING

where s is a complex number . The definit ion of a -8 offers no difficulties. I n fact , b y (3.16)

and Corollary 3.18 m

a(Z~ + i~) = a(~) H (i + ~,~ (~, ~)), 1

where 2k(~, ~) is real when ~: is real and • 0). Hence

arg a(~ +/~/) = arg a(~/) + ~ arg (i + 2~ 01, ~)),

(where arg i = Jr/2), is cont inuous and single-valued on Re Z • iF once arg a ( • is fixed.

The following es t imates are immedia te

Re 8 0 l + < I (4.14)

Re s < o ~ l a(~ +/~)-~1 < e~ + i~)l -R~

where c is a constant .

4.15. L ~ M MA. Let a E Hyp (0). Then the distributions Es(x) = Es(a, 0, x) defined by (4.13)

are entire analytic in s and

(4.16) ). > 0 =* Es(~tx ) = 2ms-nEs(x)

(4.17) S(E~) c K(A, 0)

(4.18) a(D) Es(x) = Es_l(x)

(4.19) s = 0, - -1, - 2 , ... ~ Es(x) = a(D)-S~(x).

2~ote. (4.13) should be t aken in the sense t h a t

(4.13') (E~, ~) = (2~t)-~fa(~ +i~) -~ ~g(~ +i~)d~,

where gECo(ReZ" ). T h a t the dis tr ibut ions (4.13) are analyt ic in s means t h a t all (Es, ~)

have t h a t proper ty .

Proo/. Theorem 2.4 and simple verif ications using (4.14).

2Vote. The idea of imbedding the fundamen ta l solution E = E(a, x) in an analyt ic

fami ly Esis due to M. Riesz (1949) who gives explicit formulas for Es in case a(D) is the wave

opera to r A(~]~x) = (~/~Xl) ~ - (8/~x2) 2 - . . . - (8]~xn) 2. The construct ion is also used in G~rding

(1947), Schwartz (1950) and Gelfand--Shilov (1955). For la ter reference we s ta te here the

formulas of M. Riesz. Wi th a = A as above, we have a E H y p (0,2), 0 = ( 1 , 0 . . . . . 0),

F (A, 0) = {~; ~ > 0, ~ - ~[ - . . . - ~ > 0}, K (A, 0) = {x; x~ 1> 0, ~ - ~ - . . . t> 0} and, if x E

k(A, 0),


(4.20) E, (A, ~, x) = (~ - ~ - . . . - x~)s-t~/~ (n-~)/~ 2~8-1P(s) P(s + 2 - �89 n).

When Re s > - 1 - � 8 9 this defines E s as an integrable function; when Re s ~ < - 1 - � 8 9

we define E8 by analytical continuation. Note in particular that if x is inside the light-cone

K(A, v~) then

2 p < n even ~ E(AV, v~, x) = 0

(4.21) 2p >/n even ~ E(A ~, v ~, x) = const (x~ - ~ - . . . - x2~) v-�89

n odd ~ E(A ~, v ~, x) = const (4 -x~ - . . . -x~) v-�89

where p = 1, 2 . . . . .

In order to analyse the fundamental solution E(x) = E(a, v~, x) outside the wave front

surface W = W(A, ~), we shall change the chain of integration in (4.13). The following is a

heuristic outline of our procedure.

We are going to replace the constant vector field ~ - ~ in (4.13) by certain real smooth

vector fields ~-~v(~) homotopic to it. By Cauchy's, or rather Stokes', theorem, this does

not change the integral provided we replace d~ by d(~+iv(~)), ~+iv(~) stays away from

the complex hypersurface A: a(~)=0, and the exponential stays bounded, i.e. xv(~) stays

bounded from below.

We then have

(4.22) E, (x) = (2 ~)-n f a(~ + iv(~)) -s e t~(~+~v(~)) d(~ + iv(~)). J R e Z

One way to satisfy these requirements is to choose v(~) in - F ( A , #) and to keep it bounded,

but there are wider possibilities. Localizing a at ~ we have

a(~ + iv(~)) ,-, a~ (iv(~)), ~ large

and, keeping v(~) bounded, it seems possible to allow

(4.23) v (~ )e -F~(A,~) , $ large.

As we shall see in Section 6, all such choices of v are in fact permitted. The proof uses a

semi-continuity property of the cones Ff proved in Section 5: the intersection of a F~

with neighbouring cones is never much smaller than F~ itself.

Our next step is to produce vector fields v($) such that the real part -xv(~) of the

exponential in (4.22) makes the integral convergent. To this end, let X be the complex

hyperplane x$ = 0 and suppose that 0 =~x fiRe Z' is such that

Re XN P~:~O, V~#O.

This condition actually characterizes the complement of +_ W, it means that x E-• W. We

may then strengthen (4.23) to

148 1~I. F, ATIYAI~, R. BOTT AND L. G~RDING

(4.24) v(~) E --Re X N P~(A, v~), ~ large

and a small perturbation v~(~) of v respecting (4.23) makes v~(~)x >0 for large ~. But then

we are free to let ve(~) become large for large ~, so that

v~(~)x>c]~l, c>0 , large ~.

With such a choice, there is absolute convergence in (4.22) also when x is replaced by some

near-by complex argument. This shows that Es(a, x) is holomorphic outside W.

We also want to get the explicit formulas for E(x) = E(a, v ~, x) given in the introduction.

To do this we exploit the homogeneity of a by making v~(~) positive homogeneous. Then

we have convergence difficulties at the origin, but they can be avoided by restricting s

to the half-plane Re S <0. Performing a radial integration and using formula (1.7) we then

obtain a formula for E~(x) when Re s <0 that extends by analytic continuation to s = 1.

The formula for E(x)= El(x ) at this step still involves the small parameter ~. Using the

boundary values introduced in Section 1 we now let ~-~0. The formula for E(x) tha t we

get unfortunately still involves a logarithm which we eliminate by the following device.

Restricting x to be outside - K , we have E ( - x ) = 0. Subtracting the integral formula for

E ( - x ) from that for E(x) and using (1.15) we get rid of the logarithmic term and end up,

finally, with the rational integrals (4) and (5) described in the introduction. The non-

homogeneous case is taken care of by (4.5).

To carry out this programme, we have to construct vector fields satisfying (4.23) and

(4.24) and study their homotopy properties. This will be done in Section 6 while Section

5 is devoted to the semi-continuity of the local cones F~. The actual computations sketched

above will be done in Section 7.

5. The geometry of hyperbolic surfaces. Semi-continuity of the local cones

Let a E Hyp (v ~) and A: a(~)= 0 the associated hyperbolic surface. Lemma 3.27 shows

that the real singularities of a hyperbolic surface (considered in real projective space)

are of a specially simple type. Any twodimensional plane through v ~ cuts the surface in an

algebraic curve whose singularities are multiple crossings (not necessarily transversal) of

simple branches (see figure 5a).

Fig. 5a


Consider the localization a~ of a at ~ ERe Z. The associated hypersurface A~: a~(~) : 0 ,

in other words the tangent cone of A at ~, is then also hyperbolic. By the definition of

F(A~, ~), for any U EF~ and any real point ~, the equation a~ (~+ t~ ) :0 has p real roots,

:p=m~(a) being the degree of a~. Since, near the point ~, the surface A is approximated

by its tangent cone A~, it is plausible that the equation a(~ +~ +t~) =0 will havep small real

roots for small real ~ and that this should continue to hold for every hyperbolic b close

to a. This is proved in the following lemma.

5.1. LEPTA. Suppose that a, bEHyp (vq, m), ~ E R e Z and let M be a compact part o/

1~ (A, ~). Then

(5.2) ~EM, ~EReZ, Im t :~0~ b(~+~+t~) ~:0

provided t, ~ are suHieiently small and b is su//iciently close to a.

Proo/. We normalize b and a so that b(~)=a(v a) = 1 and measure the distance from b

to a by the maximal difference [ a - b I of corresponding coefficients. Let us factor b as

follows, not exhibiting the dependence on ~ and ~,

m

(5.3) b(~ + ~ + sa) = 1-I (s + g~ (~, b)). 1

The first p = m~(a) coefficients of this polynomial,

b(~ + $ + sv a) = ~ /~ (~, b) s z + )t (~, b) s v + . . . . 0

vanish when b = a and ~ = 0 while Iv (0, a) = a~ (v~) # 0. I t follows that if [~ I and ]b - a I

are sufficiently small, then p of the numbers #k in (5.3) are small while the others are

bounded away from the origin. In the sequel let [EI and I b - a [ be small and let -/Xl ..... - # v

be the small zeros of (5.3). We shall relate #1 ..... /~v to the numbers/ul ~ .. . . ,/x ~ defined by

factorizing a~, p

a~ (~ + s~) = a~ (v ~) l-[ (s + ~u~ (~)). 1

In the first place, since the zeros of a polynomial are (multivalued) continuous functions

of the coefficients, with a suitable labelling we have

(5.4) /xk(~, b)--~uk(~, a) = oh(l), 1 <.k<~m,

where o~(1) tends to zero as Ib-a l~O, uniformly for small ~. For the same reason, since

T-~a(~ +T~ +vs~) = f i (s + T-l~tk(T~, a)) f i (~s + ~Uk(T~, a)) 1 ~ + I

150 M.F. i ' r l ~ , R. BOTT AND L. Gi~_RDINO

and also ~-ra(~ + ~ +Ts~) = a~(~ +sO) + 0(~),

it follows that, with a suitable labelling,

] < k .<p *

uniformly when ] ~[ = 1. Hence, since the/t~ (~) are homogeneous of degree 1,

a) = +o(l l )-

Combinining this with (5.4) we get

(5.5) Pk(~, b) = p0 (~) + o( I~l ) +~

Now let ~EM and replace ~ in (5.3) by ~+t~ and put s=O there. The result is

m

b(~ + ~ + t~ I) --I-~ p~(~ + ty, b) I-~ pj(~ + t~ I, b). 1 r + l

Since b EHyp (0) and ~, ~ are real, by Lemma 3.27 we can arrange the labelling so that the

functions

(5.6) t~l.tj,(~ +t~, b), l <.k<.p

are differentiable for small t. On the other hand, it is clear that

(5.7) p~ =tp~ +o~(1), l~<k</o,

where o~(1)-~0 as ~ 0 , uniformly when ~?EM and t is small. Combining this with (5.5)

we have

(5.8) /tk(~+t~, b) = t(tt~ ) 1 <.k<<.p,

where ot(1)~0 as t-~0, uniformly for small ~, Ib-a] and for ~eM and o~,a(1)~0 as ~,

[ b - a ] -~ 0, uniformly for small t and for ~ e M. Now the numbers tt ~ (7) have a positive lower

bound on M and hence (5.8) shows that to every sufficiently small ~ > 0 there is a (~ = (~(e) > 0

such that every function (5.6) has at least one real zero t=t(~, ~?, b) with ]t] <e when

~ e M , [~] <(~, In-b[ <~. Hence, under the same conditions, the equation b (~+~+t~ )=0

has at least p real roots t with ]t[ <e. But we know that the number of small roots is

precisely p. Hence the lemma follows. We can now prove that the cones F~(A), considered

as functions of ~ and a are semi-continuous in the sense explained below.

5.9. L v . ~ x . Let ~, ~ q R e Z and a, b~Hyp (O,m). Then F~+~(B, 0) contains any

preassigned compact subset M o/ F ~(A, O) provided ~ is su//iciently small and b is su//iciently

close to a.

LACUI~AS FOR t~IF:ERBOLIC DIFFERENTIAL OPERATORS 151

Proo/. Let fl = m~+~ (b) be the multiplicity of ~ + ~ relative to b, so that

b(~ + ~ +sv ~) = sq(b~+~(v~) + O(s)),

where, the coefficient of s q does not vanish (since by Lemma 3.42, b~+~ EHyp (zg)). Taking

M convex and containing 7, it suffices to show that b(~ +~ +t~) vanishes precisely of order

q at t = 0 when ~ M , ~ is sufficiently small and b is sufficiently close to a. In fact, then

b~+~(.) does not vanish on M so that F~+~(B,v~)~M. We shall prove this by a slight

modification of the arguments of Lemma 3.42. By (5.3)

(5.1o) b(~+ ~+~q +sv~)=~I (s +/~k(~ + t~/, b)) f i (s + . . . ) 1 p + l

and, by the preceding lemma and the convexity of I'~(A, ~),

Im 8~>0, Im t>~O, Im (s,-I-t) > 0 ~ b(~+~+by+s~) =~0

when ~ is real, ~ belongs to M, b is sufficiently close to a and ~, s, t are sufficiently small.

Hence, under the same conditions,

Im t > 0 *Im/~,(~+t~/, b)>0, 1 4k~<lo.

As in the proof of Lemma 3.42, this shows that the functions/~k can be labelled so that

t->Fk(~ + t~, b)

are differentiable functions of t with positive derivatives. Hence

Fk(~ +t~Y, b) = Pk(~, b) +%t(1 +O(t))

with all c~>0. Hence, using (5.10), we see that b(~+~+sv~) and b(~+~+t~) vanish to the

same order when s = 0 and t = 0 respectively. But this order is q and the proof is finished.

Before proceeding further, it is convenient to introduce the concepts of inner and outer

continuity of functions T-+C~ from some topological space to conical sets C~ in R ". We

say that such a function is inner continuous if given any T 0 and a closed conical subset

N of C~. U {0}, then C~DN = N - { 0 ) , if T is close enough to v0. By the preceding lemma, the

function a, ~-~ Fg(A, v~) is inner continuous when a e Hyp (#, m), ~ E Re Z. Outer continuity

is defined analogously: given any T 0 and any open conical set N containing 0,,, then 0 , c N

when v is close enough to T 0. Here 0 = C - {0). We now have

5.11. COROLIJARY. The local propagation cones K~(A, v~) are outer coutinuou8 /unctions

o / ~ e R e Z , a e H y p (v~, m).

152 M. F. ATI~AtI, R. BOTT AND L. G~RDING

Proo/. If C is a closed convex proper cone and T is an open conical neighbourhood of

~, then its dual T ' = {x; x ~ Re Z', xT/> 0} is a closed subset o f /} U {0} where B is the dual

of C.

Conical algebraic hypersur/aces and their duals. To provide some background for the

wave front surface we shall first deal with some generalities about conical surfaces.

When a # 0 is a homogeneous polynomial, let A be the complex hypersurface a($)=0.

Localizing a at ~ we get a new hypersurface A~: a~(~)=0, the tangent cone of A at ~. The

corresponding lineality L~=L~(A)=L(A~) is called the tangent edge of A at ~. Let m be

the degree of a and let p =m~(a). Then

a(~ + s~) = sPa~(~) + 0(8 p+I)

so that, by the homogeneity of a, aag =~tm-Pag when ~ #0. Hence A s only depends on the

ray ~ . I t is obvious that L$DL=L(A) with equality when ~EL so that ag=a. Moreover,

ELg for all ~. In fact, this is the last statement of Lemma 3.52 and its proof only uses the

homogeneity of a.

Examples. When a(~) =~0, A s is empty and L~ =Z. When a(~) =0, grad a(~) #0, then

A~=L~ is the hyperplane grad a(~)$=0. When a(~)=0 and, locally at ~,

(5.12) a ( ~ ) = h ( ~ F ' . . . h ( ~ F "

is a product of holomorphic factors vanishing at ~ but with non-vanishing gradients there,

then A s is the union of the hyperplanes grad/~(~) $ = 0 while L~ is their intersection.

As usual in agebraie geometry, a point ~EA is said to be regular if, close to ~, A has an

equation/(~) = 0 where / is holomorphie and grad/(~) #0. Any part of A consisting of regular

points is said to be regular.

The real part of A, defined as the set of real ~ such that a(~)=0 will be denoted by

Re A. A point ~ E Re A is said to be regular if it is regular in A and parts of Re A con-

sisting of regular points are said to be regular.

An xEZ' is said to be normal to A at ~ if xL~(A)=0. The corresponding hyperplane

X: x~=0, is then tangent to A at ~, i.e. XDL~(A). All x normal to A at ~ constitute the

normal of A at ~:, i.e. the orthogonal complement L~(A) in Z' of the lineality L~(A). The

normal at ~ = 0 is the orthogonal complement L~ of L(A).

Examples. When a(~)#0, the normal at ~ vanishes. When a(~)=0 but grad a(~)40,

the normal at ~ is spanned by grad a(~). When a has the faetorization (5.12) at $, the normal

at ~ is simply the linear span of all grad/q(~). When ~EA is a regular point, the normal

L A C U N A S F O R H Y P E R B O L I C D I F F E R E N T I A L OP]~RATORS 153

at ~ is a complex line, but this may also happen at singular points, e.g. if in (5.12) all

grad/q(~) are proportional.

All normals of A at points ~ : 0 constitute the dual surface ~ of A. Note that

~ cL~ In fact, L~(A)~L(A) for all ~. Since L~(A)=L(A) when ~EL(A), ~ coincides

with L~ if L(A) =4=0. In general, ~ is a rather complicated object which we shall not

analyse much further except for proving

PROPOSITIO)r. The dual ~ o /A is contained in a proper conical subvariety.

_Note. With a slightly modified definition of ~ viz. as the union of normals L~(A) for

~EL(A), ~ is contained in a proper conical subvariety of L~ This follows fl'om the

proposition by a passage from Z to the quotient Z/L(A).

Proo]. Let A k ~ A be the set of points of multiplicity ~> k. Thus

~EAk<:~ (~/~)aa(~) = 0 for 1~[ <k,

showing that Ak is a closed subvariety of Z. Hence Bk=A k -A~+ 1 is a Zariski open set on

Ak and so is a locally closed subvariety of Z. For ~ E Bk, the local lineality Lg(A) is the space

of all ~ such that ( ~ . ~ / ~ ) ~. (x!)-l(~/~)aa(~)~----0 in ~.

This shows that the subset Bk. s of all ~ E Bk such that n~(a)= s is a locally closed algebraic

subvariety and the spaces L~(A) then form an algebraic vector bundle of fibre dimension

n - s over Bk. I and also over the image B*. s of/~k.8 in projective space Z* =Z/O. The spaces

L~ (A) will then form an algebraic vector bundle over B~. s of fibre dimension s. The Zariski

closure of the set 0~.~ = UL~ (A), ~ B*.,,

is therefore an algebraic subvariety of Z' of dimension ~s +dim B~.s. Now, from the de-

finition of ~ we easily see that

oA =UCk.8, l <~k<~m, l <~s<n-1.

In fact, when L~(A)=#O, i.e. L~(A)=~Z, and ~ 4 0 then a(~)=O so that 1 <~m~(a)<~m and,

since ~EL~(A), 1 <~ng(a)~<n-1. Hence, to prove the proposition it only remains to show

that, for all values of k, s being considered,

s + d i m B~. s <~ n - 1 .

Suppose therefore that d=d(k, s )=d im Bk.s and let ~ be a non-singular point of Bk.s.

154 M. F. ATIYAH~ R. BOTT AND L. GA~RDING

Then we can choose local analytic coordinates (u, v) forZ centered at ~ so that Bk.8 is given

locally by the equations v-~0; here u = ( u 1 .. . . , ud) and v=(v 1 . . . . vn-~). Le t / (u , v) be the

analytic function which represents the polynomial a in these coordinates. Then the multi-

plicity of / at any point ~ in the domain of these coordinates is equal to m~(a) and hence,

for all ~ ~ Bk.~, it is equal to k. In particular, when u is small and fixed, / vanishes of order

1r at v=O. This shows that ~ has an expansion around u = v = O of the fo rm/= /~+~+~T. . .

where ~ is homogeneous of degree ~ and

is independent of u. Now /k corresponds to the homogeneous polynomial a~ except

for a linear change of variables. Hence a~ depends on <~n-d variables. In other words

s=n~<--.n-d and so s + dim .B*. s = s + d - l <~ n - 1

as required.

When a is a real polynomial apart from a possible complex constant factor, and ~ is

reM, the tangent edge L~(A) and the normal L~(A) are the complexifications of their real

parts Re L~(A) and Re L~(A). We shall be interested in the real dual ~ A of Re A defined

as the union of the reM normals Re Z~(A) for ~ real and ~: 0. I t is clear that ORe A ~ Re ~

and hence, by the previous proposition, ~ A is contained in a proper algebraic subvarioty

of Z'. In other words, ore A has codimension ~> 1 everywhere.

The wave/rent sur/ace. When aEHyp (v~) and ~ is real, we know from Lemma 3.42

that a~EHyp (v ~) and we have defined the local cones F~=Ff(A, v~) =F(A~, v~) and their

duals K~=K~(A,v~)=K(Avv~ ). We know that

(5.13) F ~ F = F(A, v~), K~= K = K(A, v~);

when ~GRe L(A), there is equality. Also, by Lemma 3.52

so that xEK~ :~ Re X ~ R e L ~

(5.14) K~(A ,O)c~ A, Y real ~.

Examples. When a(~)=~0, then F ~ = R e L ~ = R e Z , K~={0}. When a(~)=0, but grad

a(~)=~0, then F~ is the half-space (grad a(~).#)-l(grad a(~).~)>0 while Kf is the non-

negative span of (grad a(~).z~) -1 grad a(~), i.e. one-half of the real normal. If a is strongly

hyperbolic or, more generally, Re A has at most one-dimensional normals, these exhaust

all possibilities. When a(~)/a(~) has the factorization (5.12) at ~ with real and homogeneous

LACUNAS I~OR HYPERBOLIC DIFFERENTIAL OPERATORS 155

factors, then l~ is the intersection of the half-spaces (grad/s(~).~$)-l(grad/s(~)'~) >0

while K~ is the non-negative linear span of the corresponding real normals (grad/~(~) .v~)-~

grad/,(~).

We now come to the wave front surface.

5.15. De/inition. When a E H y p (~), let

W(A,~) = UK~(A,v~), ~ e E e 2

be the wave front surface.

I t follows from (5.13) tha t W(A, O)cK(A, v~) with equality if L(A) q:0, i.e. is the

polynomial a is not complete. In view os the definition of K~(A, v~), x E W(A, ~) if and

only if xr~(A, v ~) ~0 for at least one ~ERe 2. Hence

(5.16) x-E • W(A, v ~) ~F~(A, ~) N Re X q:O, V~ E Re Z,

which is a useful characterization of the complement of • W(A, ~).

5.17. LE~MA. Let a E H y p (~, m).

a) The wave/rent sur/ace W= W(A, ~) is a closed subset el K=K(A , ~) and

(5.18) a K c W c K N ORe A

with equality on the right when Re A has as most one-dimensional normals or when

L(A) ~0 which case W = K c ~ A.

b) The/unction a-> W(A, ~) is outer continuons. There are operators Q in hyp" (~, m)

which are arbitrarily close to a given P E h y p (0, m) and such that W(Q, ~) meets any

given conica~ neighbourhood el a ray in W(P, 0).

Proo/. a) That W c K follows from (5.13). To prove tha t W is closed, note tha t if

x'eK~ = Kg(A, ~) for some x and ~ E Re 2, then by the outer continuity os Kg as a function

of ~, y-EK~ when y, ~/are real and sufficiently close to x and ~ respectively. Hence, since the

manifold [~] = 1 is compact, yEK~ for all real ~/=~0 when y is sufficiently close to x. Hence

W is closed. In this line of reasoning, using also the outer continuity of K~(A, ~) with respect

to a, we conclude that every x outside W= W(A, 0) has an open conical neighbourhood

which is outside W(B, ~) when b E Hyp (0, m) is sufficiently close to a. Hence, if T is an

open conical neighbourhood of W, covering Re 2 - T by a finite number of open cones, we

conclude that W(B, v~)c T when b is close enough to a. Hence W(A, ~) is an outer continuous

function of a.

156 M, F. ATIYA.H, l~. BOTT AN]) L. G~RDING

The right inclusion (5.18) follows from (5.14). To prove the left inclusion, let 0 # x e~K.

Then xI~>O, I~=F(A, #) and there exists a 0 ~ e ~ such tha t x~=0. Let 7 e F t . Then,

since F~ is convex and contains #, 8# § ( 1 - 8 ) 7 e F~ when 0 ~<8 ~ 1. Since

a(~ + t(8# + (1-8)~)) = tV(a~ (80 + (1-8)~) + O(t) ),

where p = me(a), i t follows tha t there exists a t o > 0 such tha t a(~ + to(s# + ( 1 - 8 ) 7 ) ) 4 0

when 0 ~< 8 ~< 1. Since ~ + to# e F we conclude therefore tha t $0 + to~ e F so tha t x(~ + to~) =

tox 7 > 0 and, consequently, xeK~. Hence ~ K c W. Since a~=a when ~eL=L(A), we have

W = K c ORe A when L 4=0. When L = 0, then ORe A has codimension ~> 1 in Re Z and hence

W has the same property. When Re A is regular, then every 0 # ~ e r e A has precisely one

real normal, half of it being K~. Hence W = K N ~ A in tha t case. As shown by the exam-

ples below, W m a y be smaller than K N ~ A when Re A has normals of dimension > 1.

b) We know already tha t W(a, #) is an outer continuous function of a. Let / ( t ) be a

polynomial in one variable whose zeros lie in the band ] Im t ] < c. Then, if 8 is real, the zeros

o f / ( t ) -8 / ' ( t ) he in the same band. The proof of this remark is left to the reader. Let a

be the principal par t of P e h y p (#, m) and let c(~) be a real linear form with c(#) = 0 and put

Q(8) = (1 -sc(8) Va)P(8),

where Va is differentiation along #. The principal par t of Q is

b(~) = (1 -ec(~) Va)a(~),

we have b(vq)=a(#) and, by the remark above, QEhyp (#, m). In the sequel, P is supposed

to be normalized so tha t a (# )> O. Let P~ with principal par t a~ be the localization of P

at some point ~ E R e A , let ~EF~=F~(A,# ) and consider Q(~+8~). Let p=m~(a)>O be

the multiplicity of ~. A short calculation gives

b(~ + sT) = s v-1 ((a~ (7) - ~c(7) V~ a~ (7)) s - ac(~) Vo a~ (~/) + O(sZ)),

where Ve operates on 7. Hence the polynomial s~b(~ +ST) has a (p -1 ) - fo ld zero at the

origin and one small zero

(5.19) s = 80 = ~c(~e)a~(~])-lV~a~(~]) A-O(~2).

Here, since 7 EFt, a~(7) is positive and by Lemma 3.56, V~a~(z/) is also positive. Calculating

the gradient of b gives


(Vb) (~ +sT) = s '-2(Va~(~)s - ec(~) VV~a~(v) + O(e2)),

where V operates on ~ on the right. Now choose c such tha t c(~) > 0 and let ~ > 0 be small.

Then the zero (5.19) is positive for small e and inserting its value in b gives

(~Tb) (~ § so/I) = c o ~ - l ( a ~ ( / ] ) - l V a a ~ ( ~ ) Va~(~) - VVaa~(v]) § O(e)),

where c o is a positive number independent of e. Rewriting the right side we have

(5 .20) (Vb) (~ § So~ ) = C o s ~ - l a 6 ( ~ ) (VVo log a~(~) -1 § O(e)).

Now let R(~)= R~(~) be the positive ray spanned by the first term of the parenthesis. By

virtue of Lemma 3.56, these rays constitute the interior of K~=K~(A,O) relative to

L~(A), Hence, as e ~ 0 , the positive rays spanned by the left side of (5.20) with ~ E F~ come

as close as we want to any ray in this interior. This means in particular that , as e->0,

W(B, ~) comes as close as we want to any ray in K~. Now B may not be strongly hyperbolic

but repeating the operation P-->Q used above a number of times with different e and linear

functions c(~) but now starting from Q we can construct a Q 'Ehyp ~ (v ~, m) whose coeffi-

Cients tend to those of Q=Q8 with fixed e >0. (This is the procedure used by Nuij (1969).)

But then, if ~, ~EF~ and ~ > 0 are fixed and b' denotes the principal par t of Q', it is clear

tha t Vb'(~+so~ ) tends to Vb(~§ ) as b' tends to b and this finishes the proof.

Examples. The following figures show the images in real projective two-space of some

Re A and W(A, O) when n = 3 and a E Hyp (v ~, m), m = 2, 3, 4, 6. The polynomial a is supposed

to be strongly hyperbolic except in the cases 3, 5, 6, 7, 8, 9 and in the cases 3, 5, 6, 8, 9, the

dotted lines indicate Re B and W(B, ~) for a strongly hyperbolic b E H y p (v~) close to a.

In all examples, K(A, ~) is bounded by the outer contours to the right. The general properties

of Re A and ORe A considered as algebraic curves are well known. In particular, a point of

inflexion and a double point of Re A correspond to a cusp and a double tangent respectively

in its dual ~ A. In all cases we get the dual ~ A by adding to W(A, v~) all its multiple

tangents. Segments of double tangents appear in W(A, vg) in the examples 3, 5, 7, 8, 9.

In example 5, W(A, ~) has a double tangent intersecting W(A, ~) in a point. The examples

3, 5, 6 illustrate the fact tha t while W(A, ~) is an outer continuous function of a, ORe A

does not have tha t property.

The examples 6 and 9 occur in two-dimensional crystal optics and magnetogasdynam-

ics respectively (see Courant-Hilbert 1962, I I . 599-617). The example 10 is copied from

Borovikov (1961). Here Re A is close to three intersecting ellipses.

158 M. F. ATIYAH, R. BOTT AND L. GJ~RDING

I . ~ = 2

2. m = 3

Re A W (A,O)

| @

3 . ~ = 3

4. mffi4

5. m = 4

6 . ~ f f i 4

~ . ~ i . ~ ~

7. m = 4 (@ ')

~ S J

F~. 5b

LACUNAS FOR HYPERBOLIC DIFFERENTIAL OPERATORS

Re A W (A, O)

',,.x~( '.:~Ji/ j )

%,,,. ~ J '

/ \

I

, jJ ! I

' "1 !,',. l~...._ . . . . . . . . . . . __~~-~"'=/' \

X "%,. I t

'% .t

l i ) Fig. 5b

159

8. m = 6

9. m = 6

10. m = 6


6. Vector fields and cycles

Let a E H y p (v~). Following the heuristic outline a t the end of Section 4, we shall now

construct certain maps from Re Z to Z - A homotopic to the maps ~-+~ + i T, ~ E F(A, v~).

We shall only consider Coo vector fields Re Z 9 ~ ~v(~) E Re Z which are absolutely homo-

geneous

(6.1) hER ~v(2~) = ]~[v(~).

Let U be the family of such fields topologized by uniform convergence of v and all its

derivatives when [~[ = 1. I t is convenient to define certain subfamilies of U.

6.2. Definition. Let U(A, z~) be all vE U such tha t

(6.3) v(~)eF~(A,z~), V~

and, when x is real, U(A, X, z~) all v E U such tha t

(6.4) v(~)eF~(A, z?) N Re X, V~.

Let V(A, ~) c U(A, ~) and V(A, X, ~) c U(A, X, ~) be subfamilies characterized by

(6.5) 0<t-~<l =~a(~ • itv(~) ) 40 .

Note that , since F~(A, v~)=F(A~, v ~) only depends on the double ray - ~ , (6.1) and (6.3)

are consistent. By (5.16), the right side of (6.4) is never empty when x is outside • W(A, v~).

I t is clear tha t U is linear and, since the right sides of (6.3), (6.4) are convex cones, U(A, z~)

and U(A, X, O) are also convex cones, i.e. they admit linear combinations su +t v, s ~>0,

t>~O, s + t > O .

6.6. Example. Let ~(~) be any positive and absolutely homogeneous C~176 on

Re Z, e.g. y(~) = [~[. Then v(~) = 7(~)~7 belongs to V(A, z~) ~ U(A, ~7) when ~ E F(A, v~). In

fact, (6.1), (6.3) hold and, since a(~•177 for all real t * 0 ,

(6.5) is also satisfied. I f x~=0 , then v(~)E V(A, X, zT).

Two elements v 0 and v 1 in any of our families, say T, are said to be homotopic if there

js a function [O, 1] 9s-+wsE T such tha t s, ~ws(~) is infinitely differentiable and w0=v0,

Wl=V 1. Since F~(A, ~ ) = R e Z when ~ R e A, any w ~ T is homotopie to an element of T

which vanishes outside a given conical neighbourhood of Re A. We say tha t T is open if

w ~ T and v ~ U implies tha t w + ev E T when e is sufficiently small. The important families

are of course V(A, z~) and V(A, X, z$). As shown by the following lemma, they consist of the

small elements of U(A, z$) and U(A, X, ~). The lemma is a routine consequence of Lemma

5.1 and the inner continuity of the function a, ~-*P~(A, v~).

LACUNAS FOR HYPERBOLIC DIFFERENTIA~ OPERATORS 161

6.7. LEMM~t. Let aEHyp (0). Then

a) U ( A , O) is not empty and U ( A , X, O) is not empty when x E +_ W ( A , 0). Both contain

elements that vanish outside arbitrarily small conical neighbourhoods o /Re A.

b) I /Uo(A, O) and Uo are compact parts o/U(A, O) and U respectively, then Uo(A, O) +

eUoc U(B, O) and sUo(A, O)+eUoC V(B, O) provided s >0 and e are small enough

and b 6 Hyp (v q) is su//iciently close to a.

c) The/amilies U(A, 0), V(A, O) are open and connected; the/amilies U(A, X, O) and

V(A, X, O) are connected. Any compact part o/ V(A, O) or V(A, X, ~) belongs to

V(B, O) and V(B, X, O) respectively when bEI-Iyp (0) is su//iciently close to a.

d) Let xE • W(A, 0). Any v E V(A, O) such that

(6.8) v(~) EFt(A, O) n Re X

when ~ is in some conical neighbourhood o/ Re X is homotopic in V(A, O) to a

w 6 V(A, X, O) under a homotopy that respects (6.8). There exists a v 6 V(A, O) such

that xv(~) has a given constant eign /or all ~ERe 2.

Proo]. We give a general construction of elements in U(A, 0). Let ~ ERe Z and choose

an ~ EFt(A, 0). Then, by Lemma 5.9, all positive multiples of ~ belong to F~(A, 0) when

is close enough to $ and, since F~(A, 0) only depends on the double r a y / ~ , the same holds

if ~ belongs to a small enough conical neighbourhood of /~ . Hence there is a function

0<~(~)6C(Re Z) satisfying (6.1) such that ~(~)~7 EFt(A, O) when ~ is in a small enough

conical neighbourhood of /~r while q(~)UEFt(A, 0)U {0} otherwise. Covering Re 2 by a

finite number of such neighbourhoods and adding the corresponding vector fields, we get

an element of U(A, 0). When xEq- W(A, 0), then, by (5.16), F~(A, 0) f] Re X is never empty

and, chosing r/in this set, the construction gives an element of U(A, X, 0). This proves the

first part of a). The second follows from the fact that F~(A, 0 ) = R e Z when ~ is outside

Re A. I t suffices to verify b) for vector fields restricted to, e.g., the sphere I~ l= 1. In

fact, if (6.3), (6.4), (6.5) hold for such ~, by virtue of (6.1) and the homogeneity of a, they

hold for all ~. Let M(~) and N(~) be the values of v(~) and w(~) for I~l = 1 and v 6 Uo(A, 0),

and w 6 U 0 respectively. Then M(~)c F~(A, 0) and N(~) are compact sets and the functions

~-+M(~) and ~-~N(~) are outer continuous. Normahze a and b such that a (0 )=b(0)= l .

Fix an r/with 171 = 1 and let M'07 ) c Fv(A, 0) be a compact neighbourhood of M(~). Then

M(~)+eN(~)cM'(~I) if t - r / and e > 0 are small enough and, by the inner continuity of

~, b-~F ~(B, 0), M'(~)c F ~(B, O) if ~ - ~ and b - a are small enough. Hence M(~)+eN(~)c

F~(B, 0) if ~-~/, e > 0 and b - a are small enough. Covering I~l = 1 with a finite number of 11-702906 Acts mathematica, 124. Imprim6 le 9 Avril 1970,

162 M. F. ATIYAH, R. BOTT AND L. G~kRDING

such neighbourhoods, we have M(~)+sN(~)cF$(B,t~) for all ~ with 12] =1 when s > 0

and b - a are small enough. This proves the first par~ of b). To prove the second par~, note

that , by Lemma 5.1, b(~+isM'(~l))#O and hence also b(~+is(M(~)+sN(~)))#-O when

-7 , b -a , e > 0 and s > 0 are small enough. A covering argument shows tha t the same state-

ment holds for all $ with ] ~ J = 1 when b - a, s > 0 and s > 0 are small enough and this means

tha t under the same hypothesis sUo(A, ~)+ssUoc V(B, ~). This proves t h e second par t

of b). That U(A, ~) is open follows immediately from b). I f v E V(A, ~) and U 0 is a compact

par t of U then, by b), there are positive numbers so, so, ~o such tha t v(~)+ eN(~)c F~ (A, ~)

and b(~+is(v(~)+N(~)))~=O for all ~ with =1 when O<e<~so, O<s<~s o, Ib-al <00.

On the other hand, since v E V(A, ~), a(~ + isv(~)) #0 for all ~ with [ ~ ] = 1 when so ~< s ~< 1. I t

follows tha t there is an O<sl<~so such that , under the same hypothesis, a(~+is(v(~)+

sN(~))):#O when 0~<e~<s 1. Here the arguments of a constitute a compact set and hence

there is a 0<(~1<~ 0 such tha t also b(~+is(v(~)+eN(~)))#O for all ~ with I~l =1 when

s o < s ~< 1, 0 < e ~< Sl, ] b - a I ~< ~ , But the inequality is true also when 0 < s ~< s 0 and this shows

tha t v+e Uo~ V(A, t$) when s is small enough. Hence V(A, ~) is open. Since both U(A, ~)

and U(A, X, z~) are positive cones, they are connected. I f t~vt is a homotopy in U(A, ~)

or U(A, X, ~), then by b), t~sv t is a homotopy in V(A, t~) and V(A, X, ~) respectively

when s > 0 is small enough and hence V(A, t~) and V(A, X, v~) are connected. I f Vo(A, ~) is

a compact par t of V(A, vq), then by b) there is a 0 < s 0 ~< 1 such tha t s o Vo(A, t~) ~ V(B, z$)

when b is close enough to a. But a(~+isv(~))r when s~<~s<~l, vEVo(A,~) and ]$] =1

implies the same inequality when a is replaced by b and b is close enough to a. [Hence

Vo(A, ~ V(B, ~) when b is close enough to a. The same argument shows tha t any com-

pact par t of V(A, X, z~) is contained in V(B, X, ~) when b is close enough to a. This

proves e). By Lemma 3.52, L~(A, t$) + R ~ = F ~ ( A , v ~) when ~ R e Z. Hence, if ve U(A, ~)

has the proper ty (6.8), then vt(~) = v(~) -txv(~) (x~)-l~, 0 ~< t < 1, is a homotopy in U(A, z~)

from v to w=v~e U(A, X, ~) tha t respects (6.8). I f vq V(A, 0), then by a previous argu-

ment, the homotopy t~svt(~) from sv to sw takes place in V(A, t$) when s > 0 is small

enough. Hence the first par t of d) follows. To prove the last par t take voE V(A, X , v q)

and choose u e U such tha t xu(~) has a constant :sign. :/Then, since V(A, O) is open

v = % +eu~ V(A, t~) when' s > 0 is small enough and xv(~) =~xu(~) has the sign of xu($).

This finishes the proof.

Cycles. We are going to associate to our vector fields cycles and relative cycles i n

complex projective space. I n the following, all homology and cohomology is over the

complex numbers and all homology has compact supports and all cohomology has arbi t rary

supports. Further, with Z=C ~, let Z+=Z/R+ and Z*=Z/C be the quotients of 2 by the

LACUNAS FOR HYP~ERBOLIC DIFFERENTTATJ OPERATORS 163

positive real numbers and the non-zero complex numbers respectively. In particular, Z*

is (~-l)-dimensional complex projective space, while Z § is the (2n-1)-sphere. When

B c Z , let B+ and B* be the images of ~ in Z+ and Z* respectively.

Our cycles will be oriented with the aid of the differential form on Z,

n

(6.9) ~(~) = ~ ~j~j(~), 1

where vf(~) is the right cofactor of d~j in d~=d~ 1 A ... A d~n so that d~=d~jA ~j(~). The

corresponding form ~x(~) on the hyperplane X: x~=O, (x 30), is defined by

(6.10) o~(~) = d(x~) A co~(~) + O(x~).

Now, when x is real, let fl(x) + be the (n-1)-sphere Re Z+ counted with the multiplicity

�89 and oriented by

(6.11) x ~ ( ~ ) > o .

This makes fl(x) + a cycle of (Re Z +, Re X+); its boundary

(6.12) ~fl(x) + = Re X +

is an ( n - 2)-sphere with the orientation

(6.13) wx(}) > 0

induced by (6.11) and d(x})>0.

6.14. De/inition. When aEHyp (v~), xE + W(A, ~9) is real and vE V(A, X , ~), let

(6.15) ~* = a(A, x, 0)*, ~* =~(A, x, O)*EH,_I(Z* - A * , X*)

be the homology classes of the images of fl(x)+ in Z* under the maps

(6.16) ~->~-T-iv(~), ~ERo Z.

Note. In a different form and for regular Re A, the class ,r has been used by Leray

(1962) in his work on the general Laplace transform (1.c.p. 140).

The classes a*, ~* are represented by the images a*, av-* in Z * - A * of fl(x)+ under (6.16),

oriented by (6.11). Since v(~)EX for all $, these images change their orientation on X*.

The classes

(6.17) ~a*, ~*EH,_~(X* - X * fl A*) 1 1 " - 702909. Acta mathematiea, 124. Imprim6 lo 7 Avril 1970.

164 M . F . A T I Y A H , R. BOTT AND L. G ~ R D I N O

are represented by the images ~a*, ~ * in X* - A * N X* of Re X + under (6.16), oriented

by (6.13).

By Lemma 6.7, V(A, X , v~) is one homotopy class and hence the homology classes of * - *

~ , ~ do not depend on the choice of vE V(A, X , v ~) and this justifies the notation (6.15).

Let a+, ~,+ be the images of fl(x) + in Z+ under the maps (6.16) and let ~: ~-~ - ~ be the

antipodal map Z+-~Z +. Then, by (6.11), (6.13) and the absolute homogeneity (6.1) of v,

~(~+) = ( -- 1 ) n - l a +, ~ ( ~ + ) = ( -- 1 ) n - ~ a + (6.18)

so that

(6.19)

(6.20)

Note also that

(6.21)

~ * = ( - - 1 ) ~ - l a * , ~ * = ( - - 1 ) n - l O a *

2 (~* * Z~, + ( - - 1 ) n - l a *, 2 ~ * 3 a a * + ( - 1 ) n - l ~ a *.

x-E+_K(A, ~) =. ~*=0 in Hn_I(Z $ -A * , X*).

In fact, then there exists an ~/E F(A, ~q) such that x# = 0 and, putting v(~)= [~]~/, the homo-

topy ~ - i ] ~ [ ~ - ~ t ~ - i [ ~ ] # where 0~<t,.<l, contracts av + in Z + - A + to the point ( - i [ ~ [ ~ ) + in

X +. We also note in passing that

(6.22) n even, Re • N A = O ~ aa* = 0 in H,~_2(X* - X* N A*).

This follows from (6.20) if we choose v E V(A, X , ~) equal to zero in a small conical neigh-

bourhood of Re X.

The following lemma summarizes some useful continuity properties of the homology

class a ( A , x, 0)*.

6.23. L~MMA. Let a E H y p (~), x-E+_W(A,O), vE V ( A , X , O ) . Then

* /or which wE V(A, ~) and w(~) E Re X a) the class o~(A, x, ~)* contains every cycle aw

when ~ belongs to an arbitrarily small conical neighbourhood of Re X.

b) I / b E H y p (~) and y E R e Z are close enough to a and x respectively, we have a natural

commutative diagram

H~_I (Z* - A*) -~ H._I (Z* - A*, X*) --> H n - ~ (X* - X* fi A*)

H . - I ( Z * - B * ) - > H._~(Z*-B*, Y*)~ H._~.(Y*- Y* N B*),

where q~a(A, x, 0)* = a(B, y, 0)*.


Note. As we shall see in par t II, a(A, x, 0)* ~:0 in Hn_I(Z* - A * , X*) when x E K ( A , v ~) -

W ( A , #).

Proo/. a) follows from Lemma 6.7d. The homomorphisms r and ~ of the diagram of b)

are obtained by using projection onto Y*. We indicate the details. Choose an ~ ERe Z such

tha t x~ =~0 and restrict y to a ball around x where y~ =~0 and let

p = p( y): $ _ ~ _ (y~) (y~)-l~,

be the projection on Y along ~. Further, let r = r(Y) be the trace of this projection, i.e. the

line segment ( 1 - t ) $ + t p ~ , 0 ~ t ~ l , oriented by dt>O. The corresponding maps in projec-

t ive space, defined on Z* - ~ * will be denoted by p* and r* respectively. When c* c Z* - ~ * is a

compact oriented chain, then r'c*, suitably oriented by the product orientation is a compact

chain in Z * - ~ * such tha t

at*c* ~ p 'c* - c*.

Let us also put s*c*=~c*+r*Oc*

so that, if ~c* c X*, then ~s*c* ~ Y*. The maps ~ and ~ are then induced by p* and s* respec-

tively. The verification tha t the diagram is well defined and commutat ive when y, b are

close to x, a is left to the reader. The following figure illustrates the construction.

x*

y~

A*

B*

Fig. 6

I t remains to verify tha t ~ a ( A , x, v~) * = a(B, y, v~)*. Consider the chain

c(x, y) -~ r Re ~ ,

where each point counts once and c(x, y)+ is oriented by dt Aeoz(~)>0. We shall see tha t

166 M . F . ATIYAH, R. BOTT AND L. G~RDIlqG

(6.24) c(x, y)+ = f l ( y ) + - # ( x ) + .

In fact, the points of c(x, y) are

p,~=(1-t)~+tp~=~-t(y~)(y~)-~, ~eReX, 0<t<l,

and its boundary consists of R e ~ and Re Y. Since s g n x ~ : = - t s g n y ~ and sgny~t=

(1 - t ) sgn y~, the interior points r have the property that sgn y r x~ +0. Conversely,

if r is such a point, r is a positive combination of the points on the line s-~r where

it meets Re X and Re Y respectively. Hence c(x, y)+ and ~(y)+-#(x) + have the same

carrier. Now the points of/~(y)+-fl(x) + have the multiplicity �89 (sgn y r xr and hence

(6.24) holds except for a possible factor _4-1. We Ieave it to the reader to verify that the

orientations dt A ~x(~) > 0, (~ 6 Re ~) , and (sgn y ~ - sgn x~) ~(~) > 0 coincide on c(x, y)+. Next, by definition, r + a~+ has the real projection (6.24) and its points are

(6.25) {p,~-ip,v(~)} +, ~Ec(x,y), 0~<t~<l.

Further, ~+ has the real projection fl(x) + and its points are

(6.25') �89 {~ - iv(~)} +.

The boundary of s +a~ + =a+ +r+O:r + has the real projection Re Y+, its points

{p~-ipv(~)} § ~eReZ

belong to Y+. Small continuous deformations of s+a + that carry Y+ into Y+ do not change

the class of s+g + in (Z + - A +, :Y+). Replacing the imaginary parts of (6.25), (6.25') bypv(p~) and pv(p'~) is the result of such a deformation when y is close enough to x. But, in view of

(6.24), this changes s+~ + to a+~. Since V(A, z$) is open, it containspv when y is close enough $ to x and hence ~ , e ~(A,y, v~) *. The whole argument also permits small changes of a

within the class Hyp (v ~) and this finishes the proof of b).

I t follows from (6.19) that a* depends drastically on the parity of n. We shall illustrate

this further by some examples.

Example, n even. Let us first consider the case n = 2 . Choose a basis 7', ~ of Z such

that ~' eP(A, 0) and ~"eRe X and let $=u~'+v~" define coordinates u, v in Z. Taking

v--1, we may represent Z* by the complex u-plane U with a point at infinity and then

X* is represented by the origin of U and A* by a finite number of points of Re U disjoint

from zero and infinity while F(A, v~) * is represented by the component of infinity in

Re U - A * . Choosing the vector field v(~)=0 off a small conical neighbourhood of Re A,

it follows from (6.12), (6.13), (6.20) that 2a* is represented by the boundaries of small disks


centered at the image of A* and oriented by the orientation of U multiplied by the sign of

x~ taken at their centers. All these orientations are the same if and only if X* meets r (A, ~)*,

i.e. if and only if x-E +_K(A, ~). This is now also a necessary condition for a* to vanish.

Many of the features of the case n = 2 subsist when n > 2 is even. I t follows for instance

from (6.20) and the definition of V(A, X, ~) tha t ~* is represented by a tube around Re A*,

i.e. the locus of the boundary of a small two-dimensional disk whose center moves on Re A*.

The boundary of the disk is outside A* and the disk should belong to X* when its center has

tha t property. By (6.13), the orientation changes when the center of the disk passes Re A* N

Re X*. The class ~ * is represented by a tube around Re A* N X*, suitably oriented. Typi-

cally, Re A*N X* is a set of (n-3)-dimensional ovals with certain orientations. When

A* N X* is non-singular, then as will be shown in Pa r t I I , ~a* = 0 in H,_ 2 (X* - X* fi A*)

means tha t Re A* N X* is homologous to zero in A*N X*. (6.22) provides a trivial ex-

ample. Pe t rovsky (1945) p. 349-350 gives a non-trivial example when m=4. When n=4, then A* N X* is just an algebraic curve.

.Fig. 6 a. x = 2 . Crosses indicate A*, circles 2at*. ~g* vanishes.

Example, n odd. Let us first take the case n = 3 . Let 7' , ~]" be real and span X and

let ~=u~' +v~" define coordinates in X. Taking v = 1, we m a y represent X* as a complete

complex plane U. When x-E • W(A, ~), then a is not identically zero on Re X. In fact, if

this were the case, a(~) must have the factor x~ so that , if ~ E Re X, F$(A, ~) is contained in

F~(X, ~) which is a half-space not containing Re X and this contradicts the requirement

F~(A, v~) N Re X ~=O. Hence we can choose 7 ' such tha t a(~') 40 and then A* N X* is repre-

sented by a number of finite real points in U together with a number of finite complex

conjugate pairs. Further, choosing a v(~) E V(A, X, v~) which vanishes off a small neighbour-

hood of Re A, by virtue of (6.14), (6.15), (6.20), 2a~* may be represented by twice Re U

detached from the image of Re (A*N X*) as shown in figure 6b. According to (6.15), the

orientation of 9a* is tha t induced by Re X*. Note tha t 29~* is homologous in X* - A * N X*

to tubes around the non-real par~ of A* N X* oriented by the sign of I m u. They are marked

by dotted circles in the figure. Hence, if n=3, we have

(6.26) ~a* = 0 ~ A * N X* real.

In the figure 5 b, points with this property are either outside K(A, v~) or, except in the case

8, inside a curved triangle.

168 M. F. AT1-Y'AH, R. BOTT AND L. G~RDI'NG

The main features of figure 5b subsist when n > 3 is odd. In particular, by (6.12) and

(6.20), a* and ~a* may be represented by Re Z* detached from Re A* and 2Re X* detached

from Re A*N X* respectively. In the first case, the orientation changes on X*.

Q X*

Re X*

Q Fig. 6 b. n = 3. Crosses indicate A* ~ X*, lines 2 ~ * .

I t is easy to generalize one part of (6.26) to higher dimensions. This is done by the

following theorem suggested by an example of Petrovsky (1945, p. 348).

6.27. THEOREM. Let n > l be odd and let ax be the restriction o / a E H y p (v ~) to X. I /

x-E • W ( A, O) and there is a Ox E Re X such that ax 6 Hyp ( Oz) and one o/the sets • Fr N

F~(A, v~) is never empty when ~EReX , then ~g(A, x, ~)*=0 in Hn_~(X*-X*fl A*). When

all F~(A, v ~) ~ Re X are at least hall-spaces, it su//ices that a~EHyp (v~) /or some v~zERe X.

Proo/. Since the function

Re X B ~-~(Fr v~) U - Fg(A~, v~)) N F~(A, v~)

is inner continuous, the construction used in the proof of Lemma 6.7 shows that there

exists a v E V(A, X, v~) whose restriction v~ to Re X has the property that vx(~) belongs to

one of the sets _ F~(Ax, v~)~ Re X. If SERe ,4, the sign is well-defined. More precisely,

there is a continuous absolutely homogeneous function e(~) = _ 1 defined in a conical neigh-

bourhood N c R e X of Re A N R e X such that vx(~)Ee(~)F~(Ax ,~)cReX for all ~EN.

Taking vx to be zero outside N, which is no restriction, this gives us a vector field w(~)=

e(~)v~(~) belonging to V(Ax, ~x). Now aa~*=~o~(A, x, v~) * is represented by the image in

projective space X* of the cycle 7(x) + = { ~ - iv~(~); ~ E Re ~}+ oriented by ~o~(~)>0. Since

n is odd, e%(-2) =tox(~), and hence ~ * is also represented by half the image of 7(x) + plus

its conjugate. This shows that the image in X* of 7(x)+ is the same as the image of the

cycle ~l(x)+={~-iw(~); ~e REX}+ where w=evxe V(A x, ~ ) . Now all vector fields in

V(A~, ~ ) are homotopic and hence aa* is also represented by, e.g., the cycle {~-is[~lOx;

ERe X}* where s > 0. Letting s-~ cr this cycle contracts in X * - X * 0 A* to a point and


hence a~* =0 in H~_~(X* - X * fl A*). If all F~(A, 0) fl Re X are at least half-spaces (and

this is true when n=3) , then, since F~(A~, ~ ) is an open convex cone in Re X, at least

one of the sets + F~ (Az, zSx)N Fg(A, v~) is never empty. This finishes the proof.

7. Fundamental solutions expressed as rational integrals

We now have all the necessary tools for justifying our heuristic analysis at the end of

Section 4 of the formula (4.13), i.e. in integrated form

(7.1) (E~, ~) = (2 z~)-n f a(~ § i~?)-8~g(~ § i~) d(~ § i~),

where g ECo(Re Z'), ~ (x )=g( -x ) . We are going to modify the chain of the integration in

(7.1), replacing ~ by a real vector field w(~). Later we shall be able to choose w in V(A, ~).

To begin with, for reasons explained earlier, we have to work with bounded vector fields.

For this reason we introduce a cut-off operation v-~vr defined for absolutely homogeneous

vector fields by

(7.2) vr(~) = min (r/7(~), 1)v(~), r > 0 ,

where 0 <7(~)EC(Re Z) satisfies (6.1). We could choose 7(~)= I~]; our more general choice

will be useful later on. Our next theorem uses the differential (n -1 ) - fo rm eo(~) defined by

(6.9). I t has the properties that

(7.3)

(7.4)

d(gfl) A ... A d(gfn) = gndfl A ... A df, § A o)(h . . . . . fn)

d(h(~)~o(~)) = (~ ~kOh[~k § nh)d~l A ... A d~n,

where g, / . . . . are functions and h(~) a holomorphic function. Note that the first term on

the right of (7.3) vanishes when/1 ..... /~ are dependent and that the right side of (7.4)

vanishes when h(~) is homogeneous of degree - n. The theorem also employs the holomorphie

functions and distributions gs, go, defined by (1.6), (1.9), (1.11), (1.14).

7.5. THEOREM. Let a E H y p (vq, m), vE V(A, vq), gECo(ReZ' ).

When Re s < 0, then

(7.6) (E 8, ~) = (2 ~z)- n f a e z a(~)-s :~g($) dE, ~ = ~ - ivr (~),

/or all r > O.

a)

170 M.F. AT~Z~, R. BOTT XND L. OXRDrrr

b) When xv(~)<~O /or all ~ E R e Z and all xES(~), the same statement holds with vr

replaced by v.

c) The ]unctions s, x-~ Es(a , ~9, x) are holomorphic when x is outside W(A, ~) and, i] x

is outside + W(A, zP), then

E 8 (a, v q, x) = (2 zt)- n i ,~- , ~ gms-n (x~) a(~) -8 o)(~) (7.7) J r (r

- (2~)-n fr(~_i Z~ (ix~) a(~) -s (�89 ~ti - m -1 log a(~)) co(~),

where ~=~- iv (~) with vE lZ(A, vq) chosen so that xv(~) < 0 / o r all ~.

d) The ]ormula (7.7) is true also when v E V(A, X, ~), provided the integrations are taken

in the distribution sense. Di//erentiation under the integration signs is permitted.

Note. The second term of (7.7) vanishes unless m s - n = 0 , 1, 2 .. . . and then it is a

polynomial. The formula (7.7) is useful only when W(A, ~ ) # K ( A , ~), i.e. when a is a

complete polynomial. In fact, E8 vanishes outside K(A, ~). When a is not complete we

can combine (7.7) with the formula (4.8).

Proo]. For simplicity of notation, introduce the differential form

0,(~ + i~) = (2zt)-"a(~ + i~) -8 ~J(~ + i~) d(~ + i~).

I t is holomorphic and hence closed, and entire analytic in s when ~E ___F= -4-F(A, z~). By

(2.3), to every M > 0 there is a e(M)< oo such that

I I <~c(M)(1 + I~ e - i ~ [ ) - ~ e a '̀~),

h(~) = m a x x~, xES(~).

(7.8)

where

Also, by (4.12)

(7.9) R e s < 0 , r -raRe8

where c(s) > 0 is locally bounded.

First, let ~ E F, put v(~) = 7(~)~ and consider the homotopy

wt(~) = (1 - t ) ~ +tvr(~), 0-<<t<l

connecting w0(~ ) =7 with the cut-off vector field vr(~). Combining (7.8) and (7.9) gives the

following estimates of 08 =08(~-iwt(~)),

LACUNAS FOR H Y P E R B O L I C D I F F E R E N T I A L OPERATORS 171

dt = 0 ~ 10~l < c~(8) (1 + I ~ l ) - ' ~ " - ~ ~ 'h(,, Id~ I

d~(~) - - 0 ~ 1081 < the same factor times ~ Idt ^ ~J(~)l,

where zj(~) is the right cofactor of d ~ in d~, so tha t d~ = d~ s A ~j(~). The function CM(S) is

locally bounded. Also, 08(~-iwt(~)) is a holomorphic (n+ 1)-form on the compact chain

~ = {~-iw,(~); ~ <~(~) <e- l ; 0 < t < l }

and hence f Os = O. Jo C a

By the estimates above, the integrals over the parts of ~Ce where ~(~) = t , e -1 tend to zero as

e-+0 and the integrals over the other parts remain absolutely convergent. Hence (7.6)

follows for all r > 0 when v(~) = 7(~)~/, ~/E F.

Next, let v(t, ~), 0~<t~<l, be a homotopy in V(A, ~) from v(0, ~)=~(~)~ to a given

v(1, ~) =v(~) and consider the corresponding cut-off homotopy vr(t, ~) from v~(0, ~) to v,(1, ~).

We are going to estimate 0~(~), ~=~-ivr(t , ~), and first of all a($) -s. Since R e s t < 0 and

(7.10) ]a(~)-el = I a(~) I - ~ 8 e ~ ~ a%

we only have to estimate arg a(~). We claim tha t

arg a(~--iv,(t, ~))

is bounded when ~ ERe Z, 0 ~<t ~< 1. Using the homogeneity of a, it suffices to consider

q~(~, t, ~) = arg a(~-i~v($)) on the product

{~(~) : 1} • { 0 < t < 1} • { 0 < e < 1}.

Since ve V(A, vq), a ($ )40 for all SERe Z and hence ~ is a continuous function, Clearly,

is bounded when ~ = 1, and ~ being the argument of a polynomial in ~ of degree ~< m,

the variation of ~v as 0 < z ~< 1 does not exceed raze. Hence ~ is in fact bounded. By virtue

of (7.10) we have the same estimates of a($) -s as before and hence

dt = 0 ~ 10s(~)l < cM(s)~(~)-mRes (1 + I~l)-%h(~ ] (7.11)

dr(~) = 0 ~ I08 (~)[ ~< the same factor times ~ I dt ^ vr

where all dependence on r and t is displayed. Also, 08 is holomorphie on the chain

{~-iv,(t , ~); ~<?(~) ~<~-~, 0~<t~<l}.

Hence our previous argument works and we have

11~ - 702906. Acta mathematica. 124. I m p r i m 6 le 9 Avri l 1970.

172 M.F. ATIYAII, R . BOTT AND L. G~/~DING

f Os(~--ivr(O,~)) = f os(~--iVr(1, ~)),

where both integrals are absolutely convergent. This proves a).

If v($)x<~O on S(~), then h(v,.(~))<~0 and the first estimate (7.11) proves that we can

let r-+ oo in (7.6). This proves b).

To prove e) and (7.7), we first note that by Lemma 6.7 d), if x-6 + W(A, ~), there is a

v(~)6 V(A,@) such that xv(~)<O for all $ 6 R e Z . Hence, by the homogeneity, xv(~)<~ -2e?(~) for some e > 0. More generally, if y 6 Re Z' is close enough to x, say l Y-x]~< ~, then

(7.12) yv($) = -2~r(~) + (Y-X)V(~) < -~y(~).

This means that the integral

= (2 ~t)- nfa(~ - iv(~))-Se ~y(~-fvc~)) d(~ - iv(~)) (7.13) Fs(Y)

is absolutely convergent when l y - x [ <6 and Re s~<0. Introducing polar coordinates,

=~U, ~,(U)= 1, using (1.6) and (7.3) and reverting to ~ again, we have

= (2Jr) -~ i . . . . fr(~)= 1Zms-n (Y~) a(~) -s o~(~), (7.14) F , (y)

where ~=~- iv(~) . Since, by !7.12),

Im y~ ~ ~?(~),

and the functions X are holomorphie in the upper half-plane, the right side of (7.14) is a

holomorphie function of 9, s when y is close to x and Z . . . . (9~) is holomorphic in s. Hence,

by (1.6), _F~(9) is holomorphie in y, s when y is close to x and s is complex except perhaps

for simple poles when m s - n = 0 , 1, 2, .... On the other hand, we shall see that

(7.15) m s - n = O , 1, 2 . . . . ~ [ Zms_n(y~)a(~)-'og(~) =0 J

which then implies tha t Fs(y) is holomorphic in s, y for all s and all y close to x. Assume this

for the moment. Multiplying (7.13) by g(-y) where g6Co(ReZ' ) and [y-x[ <~ on S(~),

we get

(Fs, ~) = (2=)-"ja(r162 dr iv(~).

By b), the right side equals (E~, ~), which we know to be entire analytic. Hence Es(a , z~, y) = Fs(y) for all y close to x and all s. Since E8 = E~(a, ~, x) vanishes outside K(A, ~) for all s,

:LACUNAS FOR H Y P E R B O L I C DIFFERENT~AT, OPERATORS 173

this shows that E, is holomorphic in s, x when x is outside W(A, 0). At the same time, we

have (7.7) when x is outside -}- W(A, ~) and ms - n # 0 , 1, 2 . . . . . To prove (7.7) in the excep-

tional cases, just apply the operation

d

to both sides. The formula (1.13) and a small computation gives the desired result. To

prove (7.15), note tha t g . . . . (x$)a(~) -s is holomorphic and homogeneous of degree - n .

Hence the integrand is a closed (n-1)- form. But then it is independent of the choice of

v in V(A, ~) and remains the same if we put, e.g. v(~) =7(~)v ~. Next, the cycles {t~ -i?(~)v~;

~,(~) = 1} being homotopic in Z - A when t is real, letting t-~0, we see that the integral

vanishes. This finishes the proof of c).

To prove d), take a family of vector fields v =v, in (7.7) such that xv~(~) tends to zero.

We may e.g. put v(~)=w(~)-e?(~)~ where wE V(A, X, v~), x ~ = l . Then, since V(A, O) is

open, vEV(A,O) when e > 0 is small enough. Then, as e-~0, the functions Zms_n(x~)=

g . . . . (x~ +ie) tend to the distribution gm,-n(X$) on the manifold ?(~)= 1. The passage to

the limit is legitimate by Lemma 1.2. The proof is finished since, clearly, (7.7) permits

differentiation under the integration sign.

Generalization o/the Herglotz-Petrovsky-Leray/ormulaz. At long last we can now prove

the formulas (4), (5) of the introduction. Our next theorem employs a tube operation tx

from X - X N A to Z - A U X. When a c X - X N A is a compact chain, let ~r be the product

{]x~] =r} • with r so small that (TscZ-A when O<~s<~r. When a is oriented, orient

at by the product of orientation w(Re x~, Im x~)>0 of the complex plane with coordinate

x~ and the orientation of a. Then t~a =a t defines a chain map from X - X N A to Z - A U X

that increases the dimension by 1 and induces a map t~: H q ( X - X N A)~Hq+I(Z-A U X)

independent of the choice of r. We let tx denote also the induced map Hq(X* - X * N A*) --->

Hq+I(Z* - A * (J X*).

7.16. T ~ O R E M . Let aEHyp (v~, m) and let E(a, x)=E(a, ~, x) be the /undamental

solution o/a(D)=a(~/i~x) with support in K=K(A, v~). Suppose that xEW(A, v~), x E - K .

Then E(a, x) is holomorphic and

(7.17) DYE(a, x) = i(2~) 1-" ~ go (ix~) ~a(~)-lw(~) J ae*

when q=m-n-[~ l>~O and

DYE(a, x) = (2 g)-"_li~o,,*, go (ix~)~ a(~)-~ ~o(~) (7.17')

when q < O. Here

174 M. F. ATIYAH, R. BOTT AND L. G/~RDING

(7.18) Z~ g ~

and a* = o~(A, x, ~)* EHn_I(Z* - A*, X*)

is the homology class o/Definit ion 6.14.

Note. The differential (n - 1)-forms q and ~ ' appearing in (7.17) and (7.17') are rational

homogeneous of degree zero and induce rational (n -1 ) - fo rms on the projective space Z*

(see Par t II) . By (7.4), they are closed and it is clear tha t ~v is holomorphic on Z - A and

vanishes on X, while ~ ' is holomorphic on Z - A U X . Hence, if ~, a ' are (n-1)-cycles of

( Z - A , X) and Z - A U X respectively, then j ' , ~ and S~'~' depend only on the homology

classes ~* and ~'* of ~ and a ' respectively. In particular, if ~ or a ' is homologous to zero,

the corresponding integral vanishes. Hence (6.21) combined with (7.17), (7.17') reflects

the basic fact tha t E ( x ) = 0 outside K(A, v~). As remarked in the note following Theorem

7.5, our formulas are interesting only when a is a complete polynomial.

This theorem has the following corollary, which is important in the theory of lacunas

to be exposed in the next section. Let p be an integer. A function Q(x) defined in some open

conical set M is said to have homogeneity p if Q(2x)=2VQ(x) for x E M and ~>0 . I t is

obvious tha t polynomials of negative homogeneity vanish.

7.19. COROLLA:aY. I/ , /or some x,

(7.20) an(A, x, v~) * = 0 in Hn_2(X*-X* N A*),

and bEHyp (v~, ink) is su//iciently close to a ~, k = l , 2 ... . . then E(b, .) is a polynomial o]

homogeneity m k - n in the component o/the complement o/ W(B, t~) that contains x.

7.21. Examples. When n = 2 , (7.20) is always true so tha t E(a, ~, �9 ) is locally a poly-

nomial outside W(A, t$) for every a E H y p (v~). This is of course also immediate by elementary

calculation. Let us choose coordinates such tha t v q= (1, 0) and normalize a E H y p (vq, m)

such tha t a(vq) =im. Then a(~) is the product of m purely imaginary factors ak(~)=i(~l+

~k~) and we have Ek(x)= Ek(ak, v q, x ) = H ( x 1 + ~kx2)~ ( - ~ k x 1 + x~) where H is the Heaviside

function, H(t) = 1 when t >/0 and 0 otherwise. Further, E(a, x) = (E 1 * ...) (x) is a convolution.

The wave front surface W = W(A, v ~) consists of all half-rays (~, ~2k) with Q ~>0, K =

K(A , v ~) is their convex hull and, if a is complete, m >/2 then E(a, �9 ) is a positive polynomial

of homogeneity m - 2 , in each component of K - W . In particular SE(a, . ) = K , a fact

which will be of some importance later. By (6.22), if n is even and Re X N A is empty,

E(a, t$, �9 ) is a polynomial in the component L of the complement of W tha t contains x.

This applies e.g. to a product of wave operators with different speeds of fight and L equal

LACUIgAS FOR HYPERBOLIC DIFFERENT/AL OPERATORS 175

to the smallest dual fight-cone. Any aEHyp (~) close to such a product also has the cor-

responding property.

Proo] o/the corollary. Since all positive powers of a have the same associated hyper-

surface A, it suffices to consider the case k = l . If (7.20) holds, then t~a~* = 0 in

Hn_I(Z* -A* U X*) so that, by (7.17'), all derivatives of order > m - n of E(a, �9 ) vanish at x.

Now, by Lemma 6.23, (7.20) implies the same equality with x, A replaced by y, B provided

y is close to x and bEHyp (v~, m) is close to a. Hence, for every such b, E(b, .) is a poly-

nomial of degree at most m - n close to x. But then, by analytical continuation, E(b, ,. ) is

such a polynomial in the entire component L(x, B) of the complement of W(B, ~) that con-

tains x. Since E(b, .) is homogeneous of degree m - n , the polynomial has homogeneity

m - n . This finishes the proof.

Proof o/the theorem. Since x-EW(A, ~), x E--K, we have x E+_ W(A, O) so that Theorem

7.5d) applies. Letting v E V ( A , X , ~ ) and s = l and using (1.8), differentiating (7.7)

times gives

b ~E(~) = (2 ~) -" i~j'_j~ 1 Xq (x~) ~'a(~)-la~(~)

- - (2~) -n J?~ lX0q (iZ~) ~ra(~)-I ( 2 - 1 ~ -- m -1 log a(~)) eo(~),

where ~=~-iv(~) . Now since x E - K and E vanishes outside K, the right side vanishes

if x is replaced by - x . Hence

.DYE(x) = DP E(x) - ( - 1)r

Now, by (1.16) gq(x~) - ( - 1)qgq(-x~) = 2~/~q(x~)

while, if g is replaced by g0, the corresponding expression vanishes. This gives

(7.22) DYE(x) = i(2:rt)l- h i - liq ( t q J r (x~) ~r a(~)-I r

If q~>0, then, by (1.17), a.(x~) = 2-1 sgn (x~)X~(z~)

so that DYE(x) = i(2 ~ ) l - n f r = 1 2 - 1 sgn (x~) Z~ ~Va(~)-i oJ(~).

where ~=~-iv(~). In view of the definition 6.14 of ~*, this is precisely (7.17) with $ as the

integration variable. Next, let q<0. Choose coordinates such that x=(1, 0 ..... 0). Since 12 - 702909 Acsa mathemat~ca. 124. Imprim6 le 14 Avril 1970.

176 M . F . ATIYAH~ R. BOTT AND L. G~RDI!~G

o.)(~) = - d ~ l ~ go(~ 2 . . . . . ~n) -~- 'q ld~$ A ... A d~n,

th is choice gives ~x(~)= -~ (e2 . . . . . ~n), (see (6.10)). Let us also make 7(el . . . . . ~n) inde-

pendent of ~1 when ~1 is small compared to ~(~). Then, since xv(e)= vl(~)= 0, we have

~0(~) : d~l A o)x(~ 2 . . . . . ~n), ~ = e - i v ( e )

on 7(~)=1. Since, by (1.7), ffq(X~)=~(-q-1)(X) when q<0, (7.22) gives

D~E(x) = i(2 ~)l-n i q f ( _ ~/a$l)-q-l~a($)-i cox (~2 . . . . . ~), Jo g~r

where ~ ' = {~; ~ = e- iv(e) , ~ ERe X N {~(e) = 1}}

is an (n - 2)-cycle on X - X N A oriented by w~(~) > 0. Hence, if r > 0 is small, by the defini-

tion (7.18) of g~ and Cauchy's formula in one variable, we have

DYE(x) = (2~)-nfzq (ix~) ~Va(~)-ld~ 1 A o)x (~'x . . . . . ~)

with co(Re ~, Im ~) A ~o~($2, ..., ~) > 0 on the product fl = { I~ I =r} • ~a'. Here, by the

definition of ~* and tx, fl represents tx~a*. This proves (7.17').

Note. That fundamental solutions of homogeneous elliptic operators can be expressed in

terms of 'algebraic' integrals was discovered by Fredholm (1900) whose work was extended

to hyperbolic operators by Herglotz (1926-28) and Petrovsky (1945). Petrovsky's formulas

(1.c.p. 315 and 324) result from (7.17) and (7.17') when ~=0 by taking one residue onto

(A*, X*) and two successive residues onto A* N X* respectively when a is strongly hyperbolic

and A* is regular. Petrovsky's proofs are rather complicated but have later been simplified

by Leray (1952). Leray has also extended the formalism involved to a general Laplace

transform that gives explicit formulas for fundamental solutions of certain strongly hyper-

bolic operators with analytic coefficients (Leray 1962). In the cases that he considers,

Leray employs the homology class ~*.

I t is also possible to express E(x) = E(a, O, x) in terms of the distributions

ai(~) -1 = lim a(e• 0) -1, 0 <s-~0.

Taking 0-~<~(e)EC(ge2 ) such that 7(~e)= ]2]7(~ ) when ~tER and 7(~)=0 if and only if

is proportional to a fixed ~ ~ F(A, v~), the following formula can be deduced from (4.11),

E(x) ,]

LACUI~AS FOR HYPERBOLIC DIFFERENTIAL OPERATORS 177

I t is true for all x if the product of distributions appearing on the right is interpreted

properly. Using the fact that E(x) = 0 outside K = K(A, ~), this gives

E(x) = 2-1i(27~)l-n irn-n f (~rn_n (X~) a_ (~)-1(~(7(~) -- 1) d~

when xE - K . We can also use the fact that E(a, -~ , x) vanishes outside - K and get

E(x) = (2n)-~i~-~fZ~_~ (x~) (a_ (~)-1 _ a + ( ~ ) - i ) ~(~(~) _ 1) d~

when x~ - K . Here i-mE(x) is real and a_(~) - 1 - a+ (~)-1 is purely imaginary and, separating

real and imaginary parts, we get the formulas for E(x) which, for strongly hyperbolic a

have been employed by Gelfand and Shilov (1958).

The inhomogencous case. Let P E hyp (v~, m), let a E Hyp (v ~, m) be the principal part of

P = a § b and let E = E(P, ~, x) be the fundamental solution of P with support in K(P, ~) =

K(A, ~), so that, by Theorem 4.1,

(7.23) E = ~ ( - 1)kb(D)kE(a k+l, z~, x), 0

where the series converges in the distribution sense.

7.24. T~EOREM. Outside W(A,~), the series (7.23) converges locally uni/ormly and

E(P, ~, x) is a holomorphic /unction there. I/ every E ( a k+l, ~, x) is a polynomial in some

component o/ the complement o] W(A, ~), E(P, O, x) is an entire/unction there.

Proo/. Since E vanishes outside K=K(A,~) , it suffices to investigate (7.23) when

x E K - W ( A , ~). Let c(~) be a homogeneous polynomial of degree j. We can use (7.7) to

compute c(D) E(a k+l, ~, x) simply by replacing a by a ~+1 and carrying out c(D) under the

sign of integration, using (1.8). This gives

c(D) E ( a k + l, ~ , x) : (27~)-n f v=liqZq(x~) c(~) a (~) -k -10~(~)

(2~)-nf~=iZ0 q (ix~) C(~) a(~) - k - 1 (2-1~i - m -1 log a(~)) eo(~),

where q=m(k + 1 ) - n - j , ~=~-iv(~) . Let N be a complex neighbourhood o f x which is

so small that [y$] ~:0 when yEN. Put

t a=min la(~)l, tc=maxlc(~) I, to=(l +maxly~[)

when 7(~)= 1, y e N . Then (1.11) and (1.12) and obvious estimates show that

178 ~ . F . ATIYAH, R. BOTT AND L. G~LRDING

(7.25) Ic(D) E(a ~ +1, ~, y) l < Ct~ tj~-ltcP(q - 1) -1,

where y E N, q > 1 and C depends only on m, n. Next, put

b(~) = b0(~ ) + ... +bm-l(~),

where bj(~) is homogeneous of degree ~. Developing we have

b(D)~= ~ (k) bo(D)" ... bm_l (D) "--~,

where v=(v. ..... vm_l) h a s i n t e g r a l c o m p o n e n t s a n d v o + . . . + v m _ l = l ~ , ( k ) a r e t h e m u l t i -

nomial coefficients and the degree of

c(D) = be(D) ~' ... bm_l(D) ~'-1

is ?'=0v0+ lvs+.. . +(m-1 )V m_l<k (m-1 ) and tc<G~ for some C 1. Hence, by (7.25)

when p = rain (m(k + 1) - j - n) = k + m - n > O, which is true for k sufficiently large.Bumming

over the multinomial coefficients gives a new majorant

I b(D)kE( a~+z, v~, Y) I <~ CC~r(k +m - n - 1) -1,

where yGN and C~ is independent of k. Hence the series (7.23) converges uniformly in N

and this proves the first part of the theorem. To prove the second par t note tha t ff c(D) = D I"

with Iftl =m(k + 1 ) - n then the arguments tha t lead r (7.25) show tha t

(7.26) IDgE(a ~+1, O, x) I < C~ +1,

where G 8 is a constant independent of/~. If E(a ~+1, v~, �9 ) is a polynomial Q~+t(Y) in the

component L(x, A) of the complement of W(A, v ~) tha t contains x, then this polynomial

has homogeneity q ~ = m ( k + l ) - n and hence, by (7.26),

Q~+i (Y) < < C~ +t (Yl +. . . +Y,,)~k/P(ql, + 1)

in the sense of dominating power series. As before, this gives

b(D)~'Q~+l(y) < < O~ (Yl +...Yn)~/F( k + m - n + 1)

with a new constant C a. Hence E(P, zg, .) equals an entire function in L(x, A) and this

finishes the proof.

LACUNAS FOR HYPERBOLIC DIFFERENTTAT. OPERATORS 179

Chapter 3. Sharp fronts and lacunas

We shall review some basic notions and results of the theory of lacunas and make

the connection with Petrovsky's work. For Petrovsky, a lacuna for an operator P E hyp (0)

was a component of the complement of the wave front surface W(P, vq) where the funda-

mental solution E(P, t~, �9 ) vanishes. We shall take a wider point of view suggested by L.

HSrmander and based on the notion of sharp wave front. Most of the results that we state

use topological properties of algebraic manifolds tha t will be proved in Par t II .

8. Supports and singular supports of fundamental solutions

Knowing that the fundamental solution E(P)= E(P, v q, .) of a hyperbolic operator

P E h y p (vq) vanishes outside K(P, v ~) and is holomorphie outside the wave front surface

W(P, v q) gives some but not all information about the support SE(P) and the singular

support SSE(P) of E(P). We know only that

(8.1) SE(P) c K(P, z$)

(8.2) SSE(P) c W(P, z~).

As we shall see in a moment, the two inclusions may be proper. We begin by stating some

simple cases when there is equality. The following lemma employs the notations above,

a E H y p (vq) is the principal part of P E h y p (vq), L(A)=L(P) their common lineality and

n(A) =n(P) the reduced dimension, i.e. the eodimension of L(A). An index ~ indicates the

corresponding objects for the loealizations Pf , ar

8.3. L~,MY~A. I / the reduced dimension n(P) o / P is <~ 2, in particular i/n<~2, then

(8.4) SE(P) = K(A, 0).

I / the reduced dimension n~(P) o/P~ is <<. 2/or all 0:4=~ E Re Z, in particular i / R e A is regular

or i/ n <~ 3, the~

(8.5) SSE(P) = W(A, ~).

Proo/. If n(P)=O, P = a is a non-vanishing constant and the statements are trivial.

I~lext, we shall see that it suffices to prove (8.4) for complete polynomials. In fact, choose

coordinates x = (x', x") and dual coordinates ~= (~', ~") such that ~' =0 characterizes L(A).

Then P'(~ ')=P'(~) and a'(~')=a(~) are complete polynomials in ~' and E(P, ~9, x)=

E(P', ~', x)6(x ~) so that xESE(P) if and only if x 'ESE(P ' ) and x ' = 0 . On the other hand,

xEK(A , ~) if and only if x'EK(A' , ~') and x"=0 . Hence (8.4) holds for P if and only if it

180 M. F~ ATIYAH, R. BOTT AND L. G/~RDII~G

holds for P ' . The proof of (8.4) when n = 1 is immediate and is left to the reader. Also

when n = 2 and P is complete, the proof is elementary. In fact, letting a be the principal

par t of P, we know from the examples 7.21 that SE(a) = K. Now, by Theorem 7.24, E(P, #, �9 )

is holomorphie outside W and hence, if SE(P) =~K, E(P) has to vanish in some component

of K - W . But then, by (4.7), E(a) also vanishes there which is impossible. Hence (8.4)

follows. Finally, (8.5) is a consequence of (8.4) and the Localization Theorem 4.10 tha t says

tha t SSE(P) contains all SE(P~) for ~ ERe Z. In fact, if n~(P) 42, then by (8.4) SE (P~) =

K(A~, ~). I f this is true for all 0 + ~ E R e Z , by the definition of W(A, v~), we have (8.5).

We now pass to some examples showing tha t (8.1) or both (8.1) and (8.2) m a y be

proper inclusions.

8.6. Examples. When a = A p is a power of the wave operator, a E H y p (v ~) with v~=

(1, 0, ..., 0), K = K(A, zg) = K(A, v ~) is the future light-cone x 1/> 0, x12 _ x~2 _ ... _ x~ ~> 0 and

W = W(A, 0) = W(A, v~) its boundary. Since Re A: ~ - ~ - . . . - ~ = 0, ~ 40, is regular, we

know from Lemma 8.3 tha t SSE(a)= W, and this also follows from (4.20). However, by

(4.21), if 2 p < n and n is even, E(a) vanishes in /~ so tha t SE(a )= W 4 K . The simplest

case when this happens is when p = 1, n = 4 which corresponds to the propagation of light

in free space-time. That SE(a )= W reflects the possibility of omitting sharp light-signals

and is sometimes referred to as Huygens ' principle. I f P E h y p (v ~) has principal par t a = A p

and P 4 A ~, it is probable tha t S E ( P ) = K , but if, e.g., n = 6 there are operators P = A

plus lower terms with non-constant coefficients such tha t SE(P) = W (Stellmacher (1955)).

8.7. I t is easy to construct examples where both (8.1) and (8.2) or only (8.2) are proper

inclusions. In fact, let a E t typ (zT) be a product a'(~')a"(~") where ~ = (~', ~") is a partition

of the coordinates and put E ( a ) = E(a, v~, x), E(a ' )= E(a', v~', x'), E(a") = E(a', v ~', x")

and K = K ( A , zT), W = W(A, v ~) etc. Then SE(a)=SE(a' ) • SE(a"), K = K ' • and

SSE(a) = (SSE(a') • SE(a") ) 0 (SE(a') • SSE(a"))

W = (W' • K")t) (K' • W").

In fact, W is the union of all K ~ : K ~ . • for ~ 4 0 , i.e. for ~'=~0 or ~"g=0.

Hence, if, e.g., S E ( a ' ) = S S E ( a ' ) = W ' 4 K ' we get proper inclusions in (8.1) and also in

(8.2). In fact, then S E ( a ) = W ' • SE(a")~ W ' • K" is a proper par t of K and SSE(a)~

(W' • K") 0 (W' • W") does not contain (K' - W') • W"~ W. Note tha t these circumstances

prevail when a" is replaced by anyone of its powers. Hence, for given n, both (8.1) and (8.2)

m a y be proper inclusions when P = a has arbitrarily large homogeneity. The simplest choice

we can make is to put a ' ( ~ ' ) = ~ - - . . . _~2q, a , , ( ~ ) = ~ where 4<~q<n is even and p>~0 is

arbitrary. However, note tha t by replacing a ' by a '~ with 2k>~q, we have SE(a '~) = K '

SSE(a '~) = W' so tha t (8.1), (8.2) hold with equality for high powers of a.

LAOUI~AS F O R H Y P E R B O L I C D I F F E R E N T I A L OPERATORS 181

Next we shall give an example where (8.1) holds with equality but (8.2) is a proper

inclusion. Let al(~ ) 2 = ~1 - . - - ~q with q as before and put a2(~) = 2-1~ ~n. Then a = ala ~ is

a complete polynomial in H y p (v~), v~=(1, 0, ..., 0). Here K I = K ( A 1, ~ ) = K 2 = K ( A 2, ~)

and K s = K = K(A, z~) is the cone x 1 >~ 0, 2x~ ~> x~ +.. . + x~. Further, W = W(A, ~) is the union

of W I = W(A1, ~) = K 1 and W~= W(A 2, O) =~K 2. Note tha t dim W l = q < n = d i m W~ and

tha t W 1 and W 2 intersect only at the origin. An explicit calculation of E(a) = E(al) ++ E(a~)

shows tha t S E ( a ) - K while SSE(a) consists of W~ and aK 1 :~W1. In this case, (8.1) holds

with equality but (8.2) is a proper inclusion. Again, taking 2k>~q we have SE(a~)=K 1

and an explicit calculation (or an appeal to the Localization Theorem 4.10) shows tha t

SSE(a k) = W.

8.8. "Example. (Gs (1947).) Let Re Z ' be real n~-space represented by n x n

hermitian matrices x = (xj~) and let Re Z be another copy of this space put in duality with

Re Z ' by the bilinear form (~, x~=tr~x. Interpret ~/~xjk, ]g:lc, in the usual way for a

complex variable so tha t (~/~xjk) {~, x)=~j~. Pu t a(~)= det ~ and let F be the cone of all

positive definite matrices v~ E Re Z. Then a E Hyp (v ~, n) for all v ~ E F and F(A, v ~) = I ~. The

dual cone K = K ( A , ~) characterized by t r xF ~>0 consists of all x>~0 and the wave front

surface W = W(A,~) is the boundary of K. Let E~(x) be the fundamental solution of a(~/~x) ~

with support in K. In the paper quoted it is shown tha t S(E~) consists of all x E K which are

of rank ~<p. Also, if lc>~n, E(ak,+, x) is a nonvanishing multiple of (detx) k-n. Hence

S ( E . ) = K if and only if p>~n, while, if p = n - 1 , S(E~)=SS(E~)=W and, if p < n - 1 ,

S(E~) =SS(E~)g: W. If, e.g., p = 1, then S(E1)= SS(E1) has dimension 2 n - 1 which should

be compared to the dimension n ~ - 1 of W. Also, when p < n - 1, the complement of S(E~) =

SS(E,) is connected. Similar examples have been found by Gindikin and Vajnberg (1967)

and for certain parabolic operators by Gindikin (1967).

From the examples above one gets the impression tha t raising an operator P to a high

enough power will give equality in (8.1), (8.2). This is indeed the case.

The following theorem will be proved in Par t I I .

8.9. THEOREM. Let P E h y p (~, m) and let a E H y p (~, m) be its principal part. There

are integers lco and ]c 1 depending only on m and the reduced dimension n(P), such that

(8.10)

(8.11)

when k >~ k o and lc >~ ]C 1 respectively.

SSE(P k) = W(A, O)

SE(P k) = K(A, ~)

Note tha t (8.10) is a corollary of (8.11) and that , choosing k 1 as a monotone function

of m and n = n(P), we have ko(m , n)<. k l ( m - 1 , n - 1). In fact, by the Localization The-


orem 10.4, we have SSE(P~)~SE(P~) for all ~ E R e 2 and m(P~) <m, n(P$) <n so that, by

(8.11), SE(P~)=K(A ,v ~) for all ~ and consequently also SSE(Pk)D W(A,v ~) when k~>

]cl(m - 1, n - 1). More precise estimates are also possible. I f Re A is sufficiently regular or if

certain stability requirements are imposed, (8.10) holds for k >/1 and (8.11) when m]r - n >/0.

9. Sharp fronts, lacunas, regular lacunas

Petrovsky (1945), defined lacunas for a hyperbolic operator P with fundamental solu-

tion E as components of the complement of the wave front surface W in which E vanishes.

The outside of the cone of propagation K is a lacuna, called the trivial lacuna. The lacunas

inside K are of course important when we want to study the support of E, but as a l~le,

the property that E vanishes is not stable under addition of lower terms or a passage to

variable coefficients. Following a suggestion by L. HSrmander, we shall define lacunas by

a weaker property.

Starting with a general situation, let u be a distribution defined in an open subset

0 of R ~. We may think of u as describing the movement in space-time of a general elastic

( n - 1)-dimensional medium. Let C(u) be the maximal open subset of 0 where u is a Coo-

function. The complement of C(u) is then the singular support SS(u) of u. The following

definition is motivated in the introduction.

9.1. Definition. Let L be a component of C(u). The distribution u is said to be sharp

from L at a point y E l L if u has a C~176 from L to LN M for some neighbourhood

M of y. When u is sharp at all points of ~L, then L is said to be a lacuna for u. If u vanishes

in L, L is said to be strong lacuna for u.

The requirements of this definition make sense for any open part L of O and it is

sometimes convenient to allow L to be smaller than a component of C(u). We are going to

deal with the case when u = E = E(P, v ~, �9 ) is the fundamental solution of some P Ehyp (zP).

Here E is holomorphic outside the wave front surface W = W(P, ~), but as we have seen, it

may happen that the singular support of E is a proper part of W, although this does not

happen if we raise P to a high enough power. In the sequel, components L of C W = Re Z' - W

such that E = E(P) is sharp from L at all points of ~L are also called lacunas and, more

specifically, regular lacunas. The word regular will be motivated later. When L is a lacuna

of E, we also say that it is a lacuna for P. The complement of K(P, zP) is a strong lacuna

for P called the trivial lacuna. When SS(E)= W, in particular when n < 3, all lacunas are

regular.

9.2. Examples. Let P E h y p (v~, m). When n = l , E is sharp from both sides at W = ( 0 }

and W divides the real line into two lacunas one of which is the trivial one. The other lacuna

LACUI~AS F O R H Y P E R B O L I C D I F F E R E N T T A u O P E R A T O R S 183

is strong if and only if m = 0. When n = 2 and P is complete, then, by virtue of Theorem 7.24,

P is an entire function outside W so tha t the whole complement of W consists of lacunas.

Also, there are no strong lacunas apar t from the trivial one. Consequently, if n > 2 but

.L(P) has eodimension ~< 2, then S E = S S E = W = K and there is only the trivial lacuna.

When the principal par t of P is a power of the wave operator A~EHyp (0), ~ = (1, 0, ..., 0),

then K: xl>~0 , x~ 2 2 >~ - x ~ - . . . - x n ~ u is the future light-cone and W=SSE its boundary.

When P =AT, (4.21) shows tha t E is nowhere sharp at W from K if n is odd and everywhere

sharp at W from K if n is even. Hence K is a lacuna or not for P according as n is even or

odd. By Theorem 4.1, this s tatement extends to the non-homogeneous case. Again, by

(4.21), when P = A p, K is a strong lacuna if and only if 2p < n and n is even. Later we shall

give examples, of components L of the complement of W such tha t E is sharp from L at

~L except at the origin. Such an L fails to be a lacuna. Irregular strong lacunas are to be

found in the examples 8.7 and 8.8.

There is a general criterion for sharpness tha t can be presented as follows. Let n ~>2,

y E W and assume tha t W is smooth near y and tha t its curvature has maximal rank n - 2

there. I f / (x ) = 0 is a real equation of W near y with grad/ (y) ~ 0 and/~, / jk are the com-

ponents of g rad / (y ) and grad 2 ](y) respectively, this means tha t y =~0 and tha t the quad-

ratic form c=~/jkz~z k restricted to the tangent plane ~/ jz j~-O has rank n - 2 . Let ~ be

the number of negative eigenvalues of c.

9.3. TH~ORV.M. E is eharp at y/rom the side s g n / = const i /and only i/

(9.4) (sgn/)~ = ( - 1) v.

Note tha t (9.4) is independent of the choice o f / . I f n is even, (9.4) shows tha t E is

sharp from both sides of W if u is even and non-sharp from both sides when ~ is odd. I f n

is odd, E is sharp from one side but not from the other. We shall not prove Theorem 9.3

which is well known in somewhat less general forms (see Davidova (1945), Borovikov

(1959), G~rding (1961), Leray (1962), Ludwig (1965)).

9.5. Examples. When n=2, W consists of half-rays and (9.4) is identically true as it

should be. When n - 3 , 0.4) shows tha t E is sharp or non-sharp a t a curved piece of W

according to the following figure

non-sharp 7 ~ sharp

where W* is the image of W in real projective space. As we shall see in the next section,

when n = 3, an x in K - W belongs to a lacuna if and only if A* N X* is real. All the triangles

in the figures 5 b have tha t proper ty except those of the figure 5 b 8 (dotted) where Re X*

184 M. F. A IY'AH, R. BOTT AND L. G.~kRDI'NG

must meet F* for A* N X* to be real. However, the criterion applies to the dotted triangles

of figure 5 b 8 and hence E is sharp from the corresponding conical regions except at the

origin and the cusps (actually, E is sharp at the cusps). There are other instances of the

same phenomenon. Let m, n both be even, let P = a E Hyp ~ (v~) and assume tha t every

component of Re A bounds a convex cone. This would be true if, e.g., a were the product

of m/2 wave operators with different speeds of light. Then W consists of m/2 telescoping

pieces, each bounding a convex n-dimensional cone and C W has 1 § (m/2) components.

One of them is the outer component C K and one is the inner component tha t bounds

the smallest convex cone. Provided the curvature of W has maximal rank outside the

origin, (9.4) shows tha t E is sharp from both sides of W outside the origin. In fact, ~ is

either 0 or n - 2. Further, E is trivially sharp at the origin from C K and, by the examples

7.21, also from the inner component of C W. On the other hand, it has been shown by

Borovikov (1961) tha t E is not sharp at the origin from the other components of C W at

least when a is irreducible.

The following lemma shows that sharpness at the origin is the essential feature of

Iacunas.

9.6. LEMMA. Let P E h y p (v ~) and let a E H y p (v~) be the principal part o] P. Let L be a

component o/ K(A, ~) - W(A, z~) and suppose that E(P, ~, �9 ) is sharp/rom L at the origin.

Then L is a lacuna/or a and E(a, ~, �9 ) is a polynomial in L. I / E ( P k , O, �9 ) is sharp/rom L

at the origin /or a sequence o] operators PkEhyp (~) with principal parts a ~, /c=l , 2 ... . .

then L is a lacuna/or any Q E hyp (v ~) with principal part a ~.

Proo]. Let P E h y p (v ~, m). By Theorem 4.1, if t r 0, then

tn-m(~/~x)~ E(P, ~, tx) ~ (~/~x) ~ E(a, ~, x)

in the distribution sense for every ~. Now, since E(P, v~, �9 ) is sharp from L at the

origin, (~/~tx)~E(P, ~, tx) tends to a limit as t 4 0 for all v when xEL. I t follows tha t

(~/~x)VE(a, v ~, x ) = 0 when ]vI > m - n and xEL so tha t E(a, ~, .) is, in fact, a polynomial

of homogeneity m - n in L and L is a lacuna for a. The same reasoning applied to P~

shows tha t E(a k, v ~, �9 ) is a polynomial in L. But then, by Theorem 7.24, E(P, v a, �9 ) is an

entire function in L. Replacing, in this argument, a by a power of a finishes the proof.

10. Stability. Petrovsky lacunas

10.1. De]inition. Let :~ be a subset of hyp (0, m). A lacuna L for P E h y p (v~, m) is said

to be stable under :~ if every Q E :~ close enough to P has a lacuna LQ such tha t LQ N L

tends to L (i.e. absorbs any compact subset of L) as Q tends to P.

LACUI~TAS F O R H Y P E R B O L I C D I F F E R E N T I A L O P E R A T O R S 185

We shall also say tha t L is stable under perturbations in :~. As a special case, :~ m a y be

all of hyp (v ~, m). Imposing stability leaves only the regular lacunas. In fact, we have

10.2. LE~MA. A lacuna/or P Ehyp (v~, m), stable under all hyperbolic perturbations must

be regular.

Proo]. Let 0 4 x E W(P) = W(P,v~). By Lemma 5.17 there are strongly hyperbolic opera-

tors Q close to P such tha t W(Q) meets an arbitrarily small conical neighbourhood of x.

Also, by Lemma 8.3, W(Q) is the singular support of E(Q, v ~, �9 ). Hence, if a lacuna L for P

is stable under all hyperbolic perturbations, it cannot meet W(P).

The Herglo tz -Pe t rovsky-Leray formulas immediately give sufficient conditions for

regular lacunas.

10.3. T~EOREM. Let aEHyp (~, m) be complete and suppose that

(10.4) ~a(A, x, v~)* = 0 in Hn_,(X*-A* n X*)

/or some x E K(A, ~) - W(A, ~). Then x belongs to a regular lacuna L/or any b E Hyp (v~, km),

( k = l , 2 . . . . ) which is close enough to a k and L is also a lacuna ]or any P E h y p (v~) whose

principal part is such a b.

Proo/. Corollary 7.19 and Theorem 7.24.

In a somewhat less general form, the condition (10.4) was invented by Pet rovsky

(1945). He considered only strongly hyperbolic operators a with non-singular A*. I t is

convenient to have a name for components L of K(A, v ~) - W ( A , v ~) having the property

(10.4) at all points. We shall call them Petrovsky lacunas for any P E hyp (0) with principal

par t a. By Lemma 6.23, Petrovsky lacunas are stable under all hyperbolic perturbations.

Petrovsky also employed the stronger condition

(10.5) a(A, x, v~) * = 0 in Hn_I(Z*-A*, X*)

which implies (10.4). According to (7.17) it also implies tha t E(a, ~, �9 ) vanishes close to x

so tha t x would belong to a regular strong lacuna for a. However, this case is illusory

since (10,5) is not true when x is inside K(A, ~). In fact, we shall prove in Par t I I tha t in

this case, ~(A, x, v~)* has a non-zero intersection number with F(A, v~) * EHn_I(Z*-X*, A*).

Note tha t by (6.21), we know (!0.5) to hold when x is outside +_K(A, v~). For this reason,

the trivial lacuna will be included in the Petrovsky lacunas.

10.6. Examples. When n=2, then 0~*=0 for all x in K(A, ~ ) - W ( A , ~) so tha t the

entire complement of W(A, ~) consists of Petrovsky lacunas. When n = 3, then by (6.26),

186 M. F. ATIYAH, R. BOTT AND L. G~RDING

~ * = 0 if and only ff A* N X* is real. In particular, all the curved triangles in the figures

5 b except those of 5 b 8 are Pet rovsky lacunas. Theorem 6.27 provides other examples

for n odd > 3. When n is even and Re X N A is empty, x belongs to a Petrovsky lacuna.

This applies in particular to the case when a = A is the wave operator and x is inside the

future light-cone.

The interest of the Pet rovsky condition (10.4) is tha t it is both necessary and sufficient

at least when a certain stability is required. To b e ~ n with we shall prove this for low

dimensions without using stability.

10.7. LEMMA. Let aEHyp (~). I /n<.3 , all lacunas/or a are Petrovsky lacunas.

Proo/. By Lemma 8.3, W = W(A, z$) is the singular support of E(a, v~, �9 ) so t h a t all

lacunas for a are regular. The cases n = 1, 2 are trivial and the only thing we have to show

is that , for n=3, a complete and x in a lacuna for a, we have ~a(A, x, v~) * =0 , i.e., by (6.26),

tha t A* N X* is real. Since a is complete, m~>2. Put t ing lu] = m - 2 in (7.17') and using

figure 6 b, if E(a, v~, �9 ) is a polynomial of degree m - 3 close to x, we get Sv Q(z) a~(z)-ldz = 0

for all polynomials Q(z) of degree ~<m-2. Here z is an inhomogeneous coordinate in X*,

ax is the restriction of a to X* and ~ consists of small circles around the nonreal par t of

A* N X* oriented by the sign of I m z. I f w, ~ are two points in this non-real part , a suitable

choice of Q gives Sv dz/ (z- w ) ( z - ~ ) :~ 0 which is a contradiction.

A complete investigation of supports, singular supports and lacunas does not seem to

be easy, but it may be fruitful to t ry to prove or disprove some simple and strong hypo-

theses which agree with known facts. We adopt the following very tentat ive

10.8. CO~JV.CTURE. Let aEHyp (v ~, m). Then

(i) all regular lacuna8 are PetrovMcy lacunaz.

(ii) SSE(a) = (JSE(a~) for 0 = ~ E R e Z .

(iii) E(a) is holomorphie outside SSE(a).

By Theorem 10.3 and Lemma 9.6, the first par t of this conjecture implies the corresponding

s ta tement for non-homogeneous operators. In view of the structure of the Herglotz-Petrov-

sky-Leray formulas, it is tempting to consider also the s ta tement

(iv) m ~>n ~ SE(a) =K(A),

but, as we have seen in Example 8.7, this s ta tement is wrong. I t m a y still be true outside

the product situation considered there.

Pet rovsky proved a weak version of (i), namely tha t if a E Hyp ~ (v~) and A* is regular

then every stable lacuna ' i s a Pet rovsky lacuna. Pet rovsky 's method was to calculate


Hn_2(A* ) for general non-singular A* and to s tudy Abelian integrals on A* as functions of a.

In Par t I I of this paper we shall prove more precise results using theorems on the coho-

mology at algebraic varieties due to Atiyah and Hodge (1955) and generalized by Grothen-

dieck (1966). We state some of them here. By Grothendieck's theorem, all rational forms on

projective space Z* with poles on A* U X* span the cohomology of Z* - A * 0 X*. Now all

such forms of degree n - 1 appear in the formula (7.17') for D VE(a ~, x) when m/r - n - [u] < 0

a n d / c = l , 2 ... . and it is easy to show tha t the kernel of the tube operation t x vanishes.

Hence, any component of CW(A, ~) which is a lacuna for all powers of a is a Petrov-

sky lacuna, lX~ote tha t a similar cohomology argument was used in a very simple case

in the proof of Lemma 10.7. When Oo:(A, x, ~)* = O, then ~(A, x, 0)* comes from an ele-

ment of Hn-I(Z*-A*) and we can apply the same argument to (7.17) with a replaced

by powers of a. The conclusion is tha t a(A, x, ~)*=0, but we know this to be true only

when x is outside ~ K ( A , vq) *. Hence, there is no non-trivial strong lacuna common to all

powers of a. In both cases, we can also use a sharper form of Grothendieck's theorem to

be proved in Par t I I , namely tha t in order to span the desired cohomology groups, we need

only forms with a bound on the order of the poles depending only on m and n. Wha t we

then get is

10.9. THEOREM. Le$ aEHyp (~, m). There are numbers Ico and lc 1 dependinq only on

m and n such that,

(i) all regular lacunas/or a ~ are Petrovslcy lacunas when k >~ Ico.

(if) The trivial lacuna is the only strong lacuna/or a ~ when k >~ k 1.

In fact, (if) follows from the above reasoning and also (i), restricted to regular lacunas.

Now (if) is precisely the s ta tement tha t SE(a ~, ~, . )=K(A , ~). I t implies Theorem 8.9 so

that , if ]c is large enough, SS(a k, zg, . ) = W(A, ~) and this means tha t a ~ has only regular

lacunas.

10.10. COROLLARY. Let aE Hyp (v~) and let L be a component o /K(A , ~) - W(A, ~).

I / t he PetrovsIcy condition (10.4) holds/or one x in L, it holds/or all x in L.

Proof. Theorem 10.3, Theorem 10.9 (i) and the definition of a Pe t rovsky lacuma.

Example. In example 8.8,/~, i.e. the set of all x > 0 , is a Pet rovsky lacuna. Tha t the

Pet rovsky condition (10.4) holds for such an x is a non-trivial fact.

We shall also prove some more direct generalizations of Petrovsky 's result tha t do

not involve powers of a, but then we have to require some regularity of A* or stability under

188 M. F. ATIYAH, R, BOTT AND L. GARDING

certain restricted perturbation families. In particular if A* is regular, all lacunas are

Petrovsky lacunas and hence also stable under all hyperbolic perturbations. The other

parts of the conjecture are also true in this case, and (iv) holds when n<~3, and may be

true for all n. In any case, when m ~>n, a has no stable strong lacunas inside K(A).

11. Remarks about systems

Let PEhypT (~), i.e. P =P(D) has coefficients in the ring of complex r • r matrices and

det P E h y p (v~), and let E = E(P) = E(P, v~, �9 ) be the corresponding fundamental solution.

By virtue of (3.3), E(P) is locally holomorphic outside W(A, v q) where a is the principal

par t of det P. More generally, every component Esk of E has the form

Ejk = Qjk(D) E(Psk),

where Qsk(D) is a polynomial, PskEhyp (v~) is a factor of det P and Psk and Qjk are relatively

prime. Hence, if ask is the principal par t of Pro, Ejk(P) is locally holomorphic outside

W(Ajk, va) c W(A, v~). The notions of sharp front, lacuna and strong lacuna extend im-

mediately to E or any Esk and whenever E(a) or E(ajk) has such an object, E and E m

respectively also have it. Also, if Psk = am has homogeneity m m and Qm has degree >

rusk - n, a lacuna for ask is a strong lacuna of Es~. For this rather trivial reason, Esk or E

may have more strong lacunas than E(am) or E(a) respectively.

Converse propositions of the type tha t lacunas for P are also lacunas for det P should

be deduced from more detailed ones, namely tha t lacunas of Ejk are also lacunas for pro.

Throwing away the indices we are lead to the following problem about scalar operators.

Let P E hyp (v ~) and let Q(D) be a polynomial. When is a lacuna L of Q(D) E(P, v a, �9 ) a lacuna

of E(P, v ~, �9 )? Taking P=a homogeneous, it is easy to see tha t it suffices to suppose that

also Q is homogeneous. Requiring tha t L be a Petrovsky lacuna for a and using Theorem

7.16, we have the following algebraic problem. Is it true tha t all rational ( n - 1)-forms on

Z*-A*UX* with poles of order 1 on A* and arbitrary order on X* and divisible by a

fixed polynomial Q span H,~_I(Z*-A*U X*)? As we shall see in Par t I I the answer is

affirmative for instance when A* is non-singular and Q # 0 is relatively prime to a. When

n<~3, a ra ther complete analysis is possible. Systems with n = 3 possessing lacunas occur

in magnetohydrodynamics and have been studied extensively by various authors (Weitz-

net (1961), Bazer and Yen (1967), (1969), Burridge (1967)).

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Received September 30, 1969

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